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Non-linearity and dimensionality in optical heating Chang, Hung Chao Mike 2015

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Non-linearity and Dimensionality in Optical Heating by Hung Chao Mike Chang  B.A.Sc. (Electrical Engineering), The University of British Columbia, 2010  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in The Faculty of Graduate and Postdoctoral Studies  (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December, 2015   Hung Chao Mike Chang, 2015 ii   Abstract   One of the most important hurdles in electron-beam technologies such as thermionic energy conversion and parallel-beam lithography is having a high-performance electron source (cathode) material. Both of these applications, directly or indirectly, would benefit from a material’s ability to be heated efficiently through localized optical heating. Similarly, the main objective of thermoelectrics research is to maintain a high temperature gradient without hindering electrical conductivity, in order to increase the energy conversion efficiency. For this, many researchers have been pursuing the development of complex crystals with a host-and-rattling compound structure to reduce thermal conductivity Recently, localized heating with a temperature rise of a few thousand Kelvins has been induced by a low-power laser beam (< 50 mW) on the side-wall of a vertically-aligned carbon nanotube (CNT) forest. Given the excellent thermal conductivity of CNTs, such localized heating is very counterintuitive, and proper understanding of this phenomenon is necessary in order to use it for applications in thermionics and thermoelectrics. Here, an analytical formulation for solving the associated non-linear inhomogeneous heat problem through a Green’s function-based approach will be introduced. The application of this formulation to bulk metals, semiconductors, and different allotropes of carbon will be discussed. In particular, a systematic investigation will be presented on the effect of the material dimensionality and non-linear dependence of thermal conductivity on temperature. It will be shown that, if thermal conductivity is assumed to be constant, the peak temperature is proportional to the linear power density up to temperatures where radiative loss becomes significant. On the other hand, if the thermal conductivity falls with temperature, a significantly higher peak temperature and temperature gradient can be achieved. Furthermore, reducing the dimensionality of a material (going from a three-dimensional to a one-dimensional form) can lead to a significant peak temperature and temperature gradient.  iii   Preface   All the analytical derivations, numerical simulations, and analysis were performed by the author under the guidance and supervision of Dr. Alireza Nojeh. Unless otherwise specified, the text and figures were prepared by the author with assistance from Dr. Alireza Nojeh.        iv  Table of Contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables ................................................................................................................................ vi List of Figures .............................................................................................................................. vii Acknowledgements ....................................................................................................................... x Dedication ..................................................................................................................................... xi Chapter 1 ....................................................................................................................................... 1  Motivation ....................................................................................................................... 1  Thermionic Electron Emission from Carbon Nanotubes ................................................ 2  Laser-induced Heating in Carbon Nanotubes ................................................................. 3  Theoretical Background on Laser-induced Heating ....................................................... 5  Overview ......................................................................................................................... 9 Chapter 2 ..................................................................................................................................... 10  Low Temperature Model .............................................................................................. 10  Analysis of the Significance of Absorption Profile ...................................................... 18  Analysis of Optical and Thermal Parameters ............................................................... 21  Conversion to True Temperature .................................................................................. 32  Loss Correction for the Laser-Induced Heating Model ................................................ 37  Associated Error for Radiative Loss ............................................................................. 39 Chapter 3 ..................................................................................................................................... 42  Bulk Specimen: Isotropic (Metal/Semiconductor) and Planar (Pyrolytic Graphite) .... 42  Carbon Allotropes – Graphitized and Amorphous ....................................................... 45  Effect of Thermal Conductivity on Temperature .......................................................... 48  Choice of Materials for Applications in Thermionic Cathodes .................................... 54 v  Chapter 4 ..................................................................................................................................... 61 References .................................................................................................................................... 63   vi   List of Tables   Table 1. List of references for the associated heat transfer models in the literature. The coordinates of the medium and the laser source are defined in Figure 2. This thesis follows the same basic approach as the works highlighted in yellow. ............................ 8 Table 2. Material properties of the listed materials. The quoted emissivity values in the table are designated to total hemispherical emissivity with the exception of germanium where only spectral emissivity at 0.65 µm and that of molten specimen are available. The absorption coefficients are derived from the extinction coefficient based on a 532-nm source. ........................................................................................................................... 43    vii   List of Figures   Figure 1. Rise in temperature at the laser irradiation site, as noted by the apparent incandescence on the sidewall of a VA-MWCNT forest. (Left and right figures courtesy of Dr. Alireza Nojeh and Dr. Parham Yaghoobi, respectively.) ............................................................ 3 Figure 2. Schematic of the model system used throughout this work. ........................................... 7 Figure 3. Two-dimensional temperature map of the sidewall of a VA-MWCNT forest under 150 mW of laser irradiation. [52] ........................................................................................... 7 Figure 4. Plot of 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0) (solid curves, left axis) and 𝑔(𝑢, 0) (dashed curves, right axis) evaluated at the origin (𝑥𝑟 = 𝑦𝑟 = 0). The thermal conductivity anisotropicity factors are 𝛼 = 𝛾 = 1 and 𝛽 = 100 with unity eccentricity, i.e. circular Gaussian beam. ............................................................................................................................. 20 Figure 5. Distribution integral at the origin, normalized with the scenario of unity eccentricity and anisotropicity (𝛼 = 𝛽 = 𝛾 = 𝑐𝑜𝑠𝜃 = 1). Left: ∫ 𝑓(𝑢, 0, 0, 0)𝜕𝑢∞0 and right: F(α, β, γ, θ). The thermal anisotropicity is swept by tuning 𝛽 while fixing 𝛼 and 𝛾 at unity. Note that the left and right panels are two independent figures. ......................... 23 Figure 6. Complete distribution integral over the reduced spatial coordinates. The distributions are plotted over 𝑥𝑟 and 𝑦𝑟 in units of 𝑤0 with 𝛽/𝛼 ratio (upper) and eccentricity (lower) swept from 1.5 to 20, as described by Eqs. (24) and (25), normalized by its peak value. Other parameters are 𝛼𝛽 = 1, 𝛾 = 1, 𝛿𝑧 = 11.1 𝜇𝑚 and 𝑤0 = 50 𝜇𝑚 (𝛿0 = 1.59). ................................................................................................................... 24 Figure 7. Normalized temperature profile (in relation to the peak point at 𝑥𝑟 = 𝑦𝑟 = 0) of complete distribution integral evaluated along the axis on the surface. When considering the effect of the absorption correction function, a penetration depth of 11.1 µm, a beam radius of 50 µm and unity 𝛾 are used. The actual values of the coefficient corresponding to each anisotropicity factor are listed in the legend. ............................ 26 Figure 8. Absolute (upper) and normalized (lower) complete distribution integral evaluated along z-axis with unity eccentricity and anisotropicity in all three axes. The laser beam radius is fixed at 50 µm with the penetration depth swept from 0.02 µm to 500 µm. ............. 29 Figure 9. Absolute (upper) and normalized (lower) complete distribution integral evaluated along z-axis with unity eccentricity and anisotropicity in all three axes. The penetration depth is fixed at 10 µm with the laser beam radius swept from 0.5 µm to 500 µm. ............... 30 viii  Figure 10. Absolute (upper) and normalized (lower) complete distribution integral evaluated along z-axis with unity eccentricity and thermal anisotropicity in the two peripheral axes while that of the respective z-axis is scaled from 0.01 to 1000. The laser beam radius and penetration depth are fixed at 50 µm and 10 µm, respectively. ................... 31 Figure 11. The temperature dependency of thermal conductivity of carbon nanotubes in the form of ensembles [99], [103], [105], [106], [108]–[110], bundles [100], [101], [111], [112], individual nanotubes [101], [107], [113]–[115], and sintered bulks [116]–[118] derived experimentally and theoretically [104], [119]–[121]. ................................................... 33 Figure 12. The temperature dependency of thermal conductivity for an individual SWCNT with varying nanotube lengths. The term involving the nanotube length is associated to phonon-boundary scattering with a phonon mean free path of 0.5 µm [107]. .............. 35 Figure 13. The relationship between linear temperature and true temperature with the analytical model from Ref. [107] for SWCNTs of different lengths, from room temperature to 6000 K. The solid curves represent the cases where a fixed thermal conductivity is used instead of the temperature-dependent model. ............................................................... 36 Figure 14. The peak temperature versus input optical power and the associated root-mean-squared error for different combination of thermal conductivities. .............................. 40 Figure 15. Comparison between analytical solutions to that of FEA at different thermal conductivities. A medium with area of 20 mm by 10 mm and a 20-mm thickness is used as the FEA geometry and the numerical integration area of radiative loss in the analytical model. An absorption coefficient of 0.09 µm-1 and unity emissivity with no reflection loss are used. ................................................................................................. 41 Figure 16. Temperature dependent thermal conductivity for different materials. The data were taken from Ref. [123]. ................................................................................................... 44 Figure 17. Peak surface temperature versus input optical power for the listed materials. The radius of the illuminated spot was … in all cases. The left vertical axis applies to the solid curves and the right vertical axis applies to the dashed curves. ........................... 45 Figure 18. Peak surface temperature versus input optical power and the associated radiative loss using the thermal conductivity of individual SWCNTs based on Pop’s formula [107] with a fill factor of 9% applied to all cases. The radius of the illuminated spot was 50 µm in all cases. The left vertical axis applies to the solid curves and the right vertical axis applies to the dashed curves. .................................................................................. 46 Figure 19. Peak surface temperature versus input optical power for different types of carbon-based targets, include one-dimensionally anisotropic, isotropic and planar. The radius of the illuminated spot was  in all cases. ....................................................................... 47 Figure 20. Peak surface temperature versus input optical power for different temperature dependencies of thermal conductivity. The thermal conductivities of these different ix  scenarios are presented in the inset. The radius of the illuminated spot was 50 µm in all cases. ............................................................................................................................. 48 Figure 21. The three artificial thermal conductivity functions of Eqs. (41), (42) and (43) from 300 K to 5000 K. ........................................................................................................... 50 Figure 22. Temperature distribution along x-axis (upper) and y-axis (lower) for different thermal anisotropicities (corresponding to different material dimensionalities) in the case of constant thermal conductivity. The radius of the illuminated spot was 50 µm in all cases. ............................................................................................................................. 52 Figure 23. Temperature distribution along x-axis (upper) and y-axis (lower) for different thermal anisotropicities (corresponding to different material dimensionalities) in the case of a first-order decreasing thermal conductivity. The radius of the illuminated spot was 50 µm in all cases. .............................................................................................................. 53 Figure 24. Temperature distribution along x-axis (upper) and y-axis (lower) for different thermal anisotropicities (corresponding to different material dimensionalities) in the case of a second-order decreasing thermal conductivity. The radius of the illuminated spot was 50 µm in all cases. ......................................................................................................... 54 Figure 25. Linear power efficiency and the corresponding comparative radiative loss for different materials over varying temperatures. The temperature, starting at roughly 300 K (as indicated by the vertical dashed line at zero radiative loss), is varied from 500 K to 3000 K in steps of 500 K. For materials with no conductivity data available at or beyond the swept temperature points, the highest temperature is used as the last point. Inset: the temperature-dependent thermal conductivity of the materials studied herein. For anisotropic materials (marked by *), the thermal conductivity is re-scaled accordingly to yield 𝛼𝛽𝛾2 = 1. The single-axis bar-plot presents a collage of the ranges of linear-power-density to surface peak temperature rise corresponding to each material over the temperature range of interest. ............................................................ 56 Figure 26. Effect of the other parameters on the effective linear power density at the surface of the medium with unity eccentricity. Each quadrant denotes a different anisotropicity scaling approach. Only one or two anisotropicity coefficients are tuned while fixing the other(s) in each quadrant. .............................................................................................. 59 Figure 27. Effect of the other parameters on the effective linear power density at the depth at which the peak values are located. The penetration depth is with respect to the laser beam waist while the depth, Zmin, is normalized to the penetration depth. ................... 60   x   Acknowledgements   I would like to express my enduring gratitude and appreciation to my thesis advisor, Prof. Alireza Nojeh, for his continuous support and guidance. It is the utmost fortune to have the opportunity to work with him who has tremendously enlightened me with his enthusiasm for scientific exploration, professional integrity, perspicacity and creativity on many research fronts. I would like to also convey my gratitude to Prof. Shahriar Mirabbasi and Prof. John Madden for being on my thesis examination committee. I would like to thank my colleagues for their helps and valuable discussions throughout the course of my research work at the Nanostructure Lab. Last but not least, I would like to express my deepest gratitude to my family and friends for their indispensable support and encouragement.    xi   Dedication   I dedicate this work to my family.     1       Chapter 1   INTRODUCTION    Motivation Heating, a physical process commonly observed in everyday life, has a multitude of significant implications in many specialized areas at the forefront of science and technology. For instance, the phenomenon of light-induced heating of a medium may appear rather dull at first glance; however, it is a rich, complex and multi-faceted effect that is ubiquitous in many fields ranging from thermotherapy [1]–[3], machining [4]–[6], deposition [7] and annealing [8], to as far as accelerator physics, energy conversion, and inertial confinement fusion [9]. An adequate model of the associated heat transfer problem is required in order to study the ramifications of the various aspects of the materials' properties in this context. With a refined semi-analytical model, this thesis addresses the effects of thermal anisotropicity and temperature-dependency of thermal conductivity, both of which have been generally overlooked, particularly in thermionics and thermoelectrics. Laser-driven electron emission via heating from carbon nanotube (CNT) forests has been proposed for next-generation electron-beam based lithography [10] and thermionic conversion [11] for harvesting solar energy. With the increasing urge to reduce the reliance on petroleum, an efficient thermionic converter may pave the way to a clean, sustainable and affordable energy 2  alternative. In the pursuit of higher efficiency in thermoelectric energy generation, many [12]–[17] have been inspired by the concept of phonon-glass electron-crystal (PG-EC) in which a compound crystal comprises guest atoms which rattle inside a cage-like host structure. This leads to a reduced phonon mean free path and thus low thermal conductivity, while the material retains a high electrical conductivity. In all these scenarios, the temperature rise and details of the temperature distribution are of key importance in the operation of the device.   Thermionic Electron Emission from Carbon Nanotubes Since their discovery, CNTs have garnered extensive interest in many research disciplines and have been recognized by many as the most promising nano-material for the coming decades, notably in computing [18], [19], memory [20]–[23], composites reinforcement [24]–[26], sensing [27]–[29], and optoelectronics [30]–[32]. Efforts have been made to explore the potential of carbon nanotubes as a thermionic emitter. By draw-twisting a bundle from an ensemble of vertically aligned multi-walled carbon nanotubes (VA-MWCNT), Liu et al. were able to frame a 2-cm branch of ethanol-densified MWCNT yarn into a V-shaped filament for which they studied the thermionic emission characteristics [33]. The subsequent study expanded to MWCNT sheets [34] and single- and double-walled CNTs in straight-bundle form [35]. Among the CNT-based thermionic works, a few have focused on individual CNTs [36], [37] and observed interesting characteristics [38], [39] that may pertain to new emission mechanisms not observed in traditional bulk electron emitters. The above studies relied on Joule heating to raise the temperature of the cathode to sufficiently high levels for thermionic emission. Alternatively, it is possible to deliver thermal energy to the cathode by optical means: by directing a laser beam to the cathode surface, the latter can be locally heated. This approach can potentially offer greater controllability through the manipulation of the beam power, profile and pulse shape. Indeed, a number of works demonstrated laser-induced thermionic electron emission from bulk refractory materials as early as in the 1960s [40]–[48], few have explored this approach for nano-structured materials. Only recently, a thermionic cathode made from a VA-MWCNT forest subjected to a focused laser beam has been reported [49]–[54]. Figure 1 illustrates a schematic of an experiment together with a photo of the localized temperature spike which results in the incandescence observed on 3  the sidewall of a forest. CNT forests have very high optical absorption in the visible spectrum [55], which makes them good candidates for optical heating. For example, a thermionic current of 100 nA was extracted from a VA-MWCNT forest by irradiation using a low-power handheld laser (~ 4 mW) [49].  Among the works on metallic thermionic electron and ion sources, attempts have been made to theoretically study the time-resolved emission due to a pulsed laser on thoriated tungsten target [47] and tantalum strips [48]. Using the temperature derived from their analytical models, these studies deduced the current using the Richardson-Dushman equation. However, in both cases, the measured emission current did not conform to the theoretical prediction conceivably due to the neglect of grey-body radiation and the assumption of a uniform optical intensity at the cathode surface. Such effects could be even more important in scenarios involving nanostructured materials. Thus, a more refined analytical model must be created to study these systems. Exploring the laser-induced thermal response of VA-MWCNTs will lead to a better understanding of the underlying physics and pave the way to expanding the material base to other nano-structured materials.   Figure 1. Rise in temperature at the laser irradiation site, as noted by the apparent incandescence on the sidewall of a VA-MWCNT forest. (Left and right figures courtesy of Dr. Alireza Nojeh and Dr. Parham Yaghoobi, respectively.)   Laser-induced Heating in Carbon Nanotubes Laser-induced heating in VA-CNT forests has been investigated by different experimental methods. Liu et al. [56] measured the temperature due to laser-induced heating by the red-shift in Raman peaks. The location of both the G- (1595 cm-1) and D- (1350 cm-1) bands of the reported 4  Raman spectrum down-shifted linearly with increasing temperature, with the shift in G-band being more prominent. It was also noted that a peak temperature as high as 2000 K was attained in vacuum, though only 700 K in air. A more systematic study was carried out by Lim et al. who measured the peak temperature by fitting the incandescence spectral signal to Planck’s blackbody radiation formula over different optical powers [57] and under different pressures and gaseous environments [58]. Concentric multi-ring specimen damage at the interface between the vacuum and the VA-MWCNT forest was observed at the intensities used (5 – 24 mW with 10 µm spot diameter). Laser heating has also been carried out on a suspended individual single-walled carbon nanotube (SWCNT) [59]. Nakamiya et al. studied the temporal thermal response at the surface of a VA-MWCNT forest subjected to pulsed Nd:YAG and KrF lasers [60], [61]. However, their experiment was performed in air, exposing the CNTs to oxidation at elevated temperatures [62] and thermal losses through convection and ambient conduction. As mentioned before, our group has performed light-induced thermionic emission experiments from VA-MWCNT forests [10], [11], [49]–[52]. In those experiments, as can be seen from Figure 1, the forest was illuminated on the side. Despite the high thermal conductivity of nanotubes, surprisingly, it was observed that the generated heat could remain localized, allowing one to reach thermionic emission temperatures and extremely high temperature gradients (Figure 3) on this otherwise conductive materials, with very low optical intensities. The explanation proposed for this "Heat Trap" effect was based on the quasi-one-dimensionality of the sample and the steep decrease of thermal conductivity with temperature, leading to a positive feedback mechanism that limits heat conduction locally. While the simple model proposed based on this mechanism provided a reasonable match for experimental findings, it is not known that this is, indeed, the heat trapping mechanism. This is to large degree due to the fact that the heat transfer model used was overly simplified. Thus, a more elaborate theoretical study of light-induced heating of CNT forests is necessary for understanding the Heat Trap mechanism properly. This theoretical work is further motivated by the fundamental question of whether dimensionality and temperature dependence of thermal conductivity could indeed have important implications in a variety of other scenarios in thermionics and thermoelectrics.  5   Theoretical Background on Laser-induced Heating Various studies have investigated the laser-heating problem numerically with the finite difference method. Balzer and Rubahn implemented a forward-time central-space method to analyze heat conduction of a thermally-thin mica medium, where the target was assumed to be much thinner than the thermal diffusion length along the optical axis, with surface radiation [63]. Others have assumed a uniform optical intensity profile over the area of interest [63]–[65] and thermally isotropic media [63]–[66]. The finite element method was used by Burgess et al. to analyze the temporal response of an optical disk with tellurium and organic dye as the optical absorbing thin film [67]. With the structural flexibility offered by numerical methods, Cavallier et al. studied the thermo-elastic displacement on the rear surface of an etched silicon grating irradiated by a modulated argon ion laser beam [68]. Numerical approaches can indeed provide more versatility and flexibility, although they suffer greatly in terms of computation time as the accuracy of the solution is susceptible to the fineness of the meshing and round-off errors. In contrast, analytical solutions provide intuition for parametric optimization and a solution to the inverse problem (extracting the material's thermal parameters from measured temperature data). Numerous proposed three-dimensional models have assumed a thermally isotopic material as the heating medium [69]–[76]. One of the approaches was to follow Duhamel’s theorem to transform the linearized inhomogeneous partial differential equation to a homogeneous initial-value problem, then derive the inhomogeneous solution from the fundamental solution to this auxiliary problem [77]. This allows a flexible energy source term particularly in the time domain for simulating arbitrary laser pulse shapes. Others have solved the inhomogeneous heat equation through direct Hankel transform for continuous-wave [75], [76] and pulsed beams [78], [79] on cylindrical coordinates. However, the cylindrical coordinates rely on a simplified source, either uniform or circular, and a thermally isotropic material. Yibas and Al-Aqeeli applied the Laplace transform to the one-dimensional heat equation with an exponentially decaying laser pulse in both the spatial and temporal domains and deduced the temporal temperature profile by Laplace inversion from the s-domain [80]. CNTs and many other nanomaterials are highly anisotropic due to their one- or two-dimensional nature. Therefore, the isotropic model is not applicable for studying VA-CNT forest. 6  Another analytical method to solve the inhomogeneous heat equation is to utilize Green’s function, as widely implemented by many [69]–[71]. The temperature distribution can be formulated by the convolution of the Green’s function and the source term as 𝑇(𝑥, 𝑦, 𝑧, 𝑡) = ∫∭𝐺(𝑥, 𝑦, 𝑧, 𝑡, 𝑥𝑠, 𝑦𝑠, 𝑧𝑠, 𝑡𝑠)𝑄0′′′(𝑥𝑠, 𝑦𝑠, 𝑧𝑠, 𝑡𝑠)𝜕𝑥𝑠𝜕𝑦𝑠𝜕𝑧𝑠𝜕𝑡𝑠 , where 𝐺(𝑥, 𝑦, 𝑧, 𝑡, 𝑥𝑠, 𝑦𝑠, 𝑧𝑠, 𝑡𝑠) is the Green’s function associated with the heat equation and 𝑄0′′′(𝑥𝑠, 𝑦𝑠, 𝑧𝑠, 𝑡𝑠) describes the input thermal energy. This provides a temporally- and spatially-resolved model for an anisotropic material with arbitrary optical inputs, while being computationally efficient as one only has to numerically evaluate the integral with the proper heat kernel and the energy source function. In light of this method, Abtahi et al., and Burgener and Reedy have modelled a dual-layer stack as the heating medium with stationary [81] and scanning [82] laser beams. Analytical models for more complex N-layer film stacks also exist in the literature [78], [83]–[85]. By using Cartesian coordinates, more versatile models, incorporating an elliptical Gaussian source [83], [85]–[89] and thermally anisotropic media [85]–[87], [90], can be derived. Sparks has particularly investigated the evolution of temperature by varying the medium thickness in relation to the penetration depth and diameter of a Gaussian source [72]. Assuming optical absorption to take place entirely on the surface, Lee and Albright carried out a sensitivity analysis aiming to provide a parametric effectiveness measure for process variations in laser surface treatment [91]. Groenbeck and Reichling [92] studied pulsed-laser-induced heat waves in an anisotropic film stacked on an isotropic substrate with constant thermal conductivities. By following the Fourier-transform method developed by Iravani and Nikoonahad for thermo-acoustic analysis [93], they were able to derive an analytical model of the frequency-dependent temperature rise in response to a modulated source. Treatment of the melting of the medium at elevated temperatures has been studied by a few [89], [94]; however, this thesis work will be limited to a temperature range where the integrity of the material studied remains uncompromised as cathode deformation is impermissible in practical applications. Table 1 summarizes the relevant literature works in regard to the different laser-induced thermal models. As can be seen from the table, the present work is the only one that includes all the following effects simultaneously: optical penetration, anisotropicity, temperature dependence of 7  thermal conductivity, radiative loss (in addition to conductive loss). These are all crucial in the analysis of heating in novel nanostructured materials such as VA-CNT forests.  Figure 2. Schematic of the model system used throughout this work.   Figure 3. Two-dimensional temperature map of the sidewall of a VA-MWCNT forest under 150 mW of laser irradiation. [52]  8  Table 1. List of references for the associated heat transfer models in the literature. The coordinates of the medium and the laser source are defined in Figure 2. This thesis follows the same basic approach as the works highlighted in yellow. Ref. 𝑄0′′′(𝑥, 𝑦, 𝑧)= 𝐼0,x,y(𝑥, 𝑦)𝐼0,z(𝑧) Aniso. k(T) Multi-layer Temporal Loss Method 𝐼0,x,y(𝑥, 𝑦) 𝐼0,z(𝑧) Scann. Pulsed Trans. This Work 𝑒−𝑥2−𝑦2  𝑒−𝑧        (rad) Green’s [94] 𝑒−𝑟2  Π(𝑧)        Green’s [75] 𝑒−𝑟2 𝑒−𝑧        Hankel Transform [76] 𝑒−𝑟2 𝑒−𝑧        Hankel Transform [86] 𝑒−𝑥2−𝑦2  δ(𝑧)        Green’s [87] 𝑒−𝑥2−𝑦2  δ(𝑧)        Green’s [84], [85] 𝑒−𝑥2−𝑦2  δ(𝑧)        Green’s [89] 𝑒−𝑥2−𝑦2  δ(𝑧)        Green’s [88] 𝑒−𝑥2−𝑦2  δ(𝑧)        Green’s [90] 𝑒−𝑟2 δ(𝑧)        Green’s [66] 𝑒−𝑟2 ,  Π(𝑟) 𝑒−𝑧     (𝑒𝜂(𝑡𝜏−1.5)2)   (rad) FDM [65] 1 Λ(𝑡)𝑑𝑆𝑑𝑧     (Λ(𝑡))   FDM [63] 𝑒−𝑟2 1        (rad) FDM-FTCS [64] 𝑒−𝑟2 δ(𝑧)     (Λ(𝑡))   FDM-FTCS [67] 𝑒−𝑟2 𝑒−𝑧     (𝑡𝑟𝑎𝑝(𝑡))   FEM [68] 𝑒−𝑟2 𝑒−𝑧     (𝑒𝑖𝑤𝑡)   (conv) FEM [81] 𝑒−𝑟2 ,  Π(𝑟) 𝑒−𝑧        Green’s [82] 𝑒−𝑟2 δ(𝑧)        Green’s [70], [71] 𝑒−𝑟2 𝑒−𝑧        (conv) Green’s [69] 𝑒−𝑟2 𝑒−𝑧        Green’s [91] 𝑒−𝑟2 δ(𝑧)        Green’s, FDM [83] 𝑒−𝑥2−𝑦2 𝑑𝑆𝑑𝑧     (Π(𝑡), 𝑡𝑟𝑎𝑝(𝑡), 𝑡𝑟𝑎𝑝(𝑡) with exp-transient)   FFT on Green’s, FDM [95] Π(𝑟) 𝑒−𝑧        Approximate solution [73] 𝑒−𝑟2 𝑒−𝑧, δ(𝑧)     (𝛿(𝑡), Π(𝑡), 𝑒−𝑡2, Q-switched)   Green’s [77] 1 𝑒−𝑧     Q-switched   Duhamel’s [96] 1 δ(𝑧)        Approximate solution [80] 1 𝑒−𝑧     (𝑒−𝑡)   Laplace Transform, FDM [74] 𝑒−𝑟2 δ(𝑧)     (𝑡𝑒−𝑡)   Duhamel’s [92] 𝑒−𝑟2 𝑒−𝑧     (𝑒𝑖𝑤𝑡)   2D Fourier Transform [78] 𝑒−𝑟2 𝑒−𝑧     (𝑒𝑖𝑤𝑡)   Hankel Transform [93] 𝑒−𝑟2 𝑒−𝑧     (𝑒𝑖𝑤𝑡)   2D Fourier Transform [79] (1 + 𝑟2)𝑒−𝑟2 δ(𝑧)     (CW, Π(𝑡))   Hankel Transform  9    Overview This thesis comprises two parts. The first section describes an analytical model for laser-induced heating on an anisotropic medium with temperature dependent thermal conductivity. It provides an overview of the existing solution by Lu [86] and describes an extended model incorporating an exponential absorption function. A correction for radiative loss is introduced and its limitation discussed. Then, finite element analysis using the heat transfer module of COMSOL Multiphysics is used to address this non-linear problem and compare with the outcome of the analytical model. The inherent errors resulting from the assumptions of the analytical model are also discussed. In the second section, the application of the analytical model to a variety of different materials is presented. This latter section will demonstrate the relevance of dimensionality and temperature dependency of the material in the heating problem, as well as provide a better understanding of the property requirements of the cathode material for an efficient laser-induced thermionic electron source.    10       Chapter 2   Analytical Model for Laser-Induced Heating     Low Temperature Model Referring to Figure 2, consider the general energy balance equation for the inhomogeneous heat problem,  −∭𝛻 ∙ ?⃑?𝜕𝑉 +∭𝑄0′′′𝜕𝑉 =∭𝜌𝐶𝑝𝜕𝑇𝜕𝑡𝜕𝑉 (1) , where 𝜌 is the mass density of the heating medium in kg/m3, 𝐶𝑝 is the specific heat in J/kgK, ?⃑? is heat flux density in W/m2, described by Fourier’s law of thermal conduction,  𝑞𝑖⃑⃑⃑ ⃑ = −∑𝑘𝑖𝑗𝛻𝑗𝑇𝑗 𝑘 = [𝑘𝑥𝑥 𝑘𝑥𝑦 𝑘𝑥𝑧𝑘𝑦𝑥 𝑘𝑦𝑦 𝑘𝑦𝑧𝑘𝑧𝑥 𝑘𝑧𝑦 𝑘𝑧𝑧] (2) 11  , where 𝑘 is the thermal conductivity tensor of a general, anisotropic medium and 𝑄0′′′ is the heat generation term in W/m3. The latter follows a Gaussian distribution given by the optical intensity of an ideal laser beam at the fundamental mode, 𝑄0′′′ = 𝐼0(1 − 𝛤)𝑒−2(𝑥−𝑥0)2𝑤𝑥2−2(𝑦−𝑦0)2𝑤𝑦2 𝑒−𝛼𝑧|𝑧| , where 𝐼0 is the peak intensity, 𝛤 is the reflection coefficient to account for any loss of optical power at the surface, wx and wy are the laser beam waists along the x- and y-axis defined at the 1/e2 point, x0 and y0 are the coordinates of the laser location, and αz is the optical absorption coefficient of the incident laser in the medium. The absorption coefficient can be found directly by optical attenuation measurement or derived from the extinction coefficient following the relation  𝛼𝑧 =4𝜋𝑘𝑒𝑥𝑡𝜆 (3) , where kext is the extinction coefficient or, equivalently, the imaginary part of the refractive index of the material, and  λ is the wavelength of the incident light. The optical absorption coefficient has been assumed to be independent of temperature and intensity. The peak volumetric power density can be derived by taking the volume integral of the heat generation term, leading to 2𝑃0𝛼𝑧𝜋𝑤𝑥𝑤𝑦 for an exponentially decaying absorption profile, and  4𝑃0𝜋𝑤𝑥𝑤𝑦 for abrupt absorption on the surface, represented by a Dirac delta function in the z direction. Placing the Gaussian beam at the x-y origin on a semi-infinite geometry will give rise to an energy source term as 12   Q0′′′ =2P0(1−Γ)αzπwxwye−2x2wx2−2y2wy2e−αzz . (4)  For isotropic materials, the conductivity tensor reduces to a single term, and is a diagonal matrix in the case of anisotropic materials (such as composites or low-dimensional materials like carbon nanotube forests), assuming that the axes of anisotropy are aligned in the x, y and z directions. The inhomogeneous heat equation for an anisotropic medium under an external Gaussian optical input is then  𝜕𝜕𝑥(𝑘𝑥𝑥𝜕𝑇𝜕𝑥) +𝜕𝜕𝑦(𝑘𝑦𝑦𝜕𝑇𝜕𝑦) +𝜕𝜕𝑧(𝑘𝑧𝑧𝜕𝑇𝜕𝑧)+2𝑃0(1 − 𝛤)𝛼𝑧𝜋𝑤𝑥𝑤𝑦𝑒−2𝑥2𝑤𝑥2−2𝑦2𝑤𝑦2𝑒−𝛼𝑧𝑧 = 𝜌𝐶𝑝(𝑇)𝜕𝑇𝜕𝑡 (5) , where kxx, kyy, kzz are the thermal conductivity components along the three axes. These components, in practice, are temperature dependent. It will be assumed that these components follow the same temperature dependence. This assumption, while presenting a limitation, will be needed in order to allow for an analytical treatment of the problem, and may or may not be fully justified for different materials. In this manner, a generalized anisotropic model is formulated as   𝜕𝜕𝑥′(𝑘(𝑇)𝜕𝑇𝜕𝑥′) +𝜕𝜕𝑦′(𝑘(𝑇)𝜕𝑇𝜕𝑦′) +𝜕𝜕𝑧′(𝑘(𝑇)𝜕𝑇𝜕𝑧′)+2𝑃0(1 − 𝛤)𝛼𝑧𝜋𝑤𝑥𝑤𝑦𝑒−2𝛼𝑥′2𝑤𝑥2−2𝛽𝑦′2𝑤𝑦2𝑒−𝛼𝑧√𝛾𝑧′= 𝜌𝐶𝑝(𝑇)𝜕𝑇𝜕𝑡  (6) with  𝛼 =𝑘𝑥𝑥𝑘(𝑇), 𝑥′ =𝑥√𝛼 (7) 13   𝛽 =𝑘𝑦𝑦𝑘(𝑇), 𝑦′ =𝑦√𝛽  𝛾 =𝑘𝑧𝑧𝑘(𝑇), 𝑧′ =𝑧√𝛾 , where 𝛼, 𝛽, 𝛾 are the thermal anisotropicity factors associated to the three spatial axes, and 𝑘(𝑇) represents the temperature dependence. Subsequent elimination of the non-linear 𝑘(𝑇) component can be done through the Kirchhoff transform [76], [86]–[89],   [−𝑘0 (𝜕2𝜕𝑥′2+𝜕2𝜕𝑦′2+𝜕2𝜕𝑧′2) +𝜌𝐶𝑝(𝑇)𝑘(𝑇)𝑘0𝜕𝜕𝑡]𝜙(𝑥′, 𝑦′, 𝑧′, 𝑡)=2𝑃0(1 − 𝛤)𝛼𝑧𝜋𝑤𝑥𝑤𝑦𝑒−2𝛼𝑥′2𝑤𝑥2−2𝛽𝑦′2𝑤𝑦2𝑒−𝛼𝑧√𝛾𝑧′ (8) with  𝜙 =1𝑘0∫𝑘(𝜏) 𝜕𝜏𝑇𝑇0, 𝑘0 = 𝑘(𝑇0) (9) , where 𝜙 is a quantity called linear temperature in Kelvin from which conversion to true temperature is achieved by the integral normalized over the room-temperature value, 𝑘0. The problem is further simplified by assuming the thermal diffusivity, 𝛼𝐷, to be temperature independent. This might sound unjustified, given that thermal diffusivity is related to thermal conductivity, for which the temperature dependence is taken into account. However, given the physical insight that, in steady state, thermal diffusivity does not play a role, this assumption may be used, albeit with caution. Indeed, further support for this assumption will be found a posteriori, using comparisons with numerical simulation results. This assumption was also implicitly used by Lu [86], following the work by Moody and Hendel [89] which showed that the diffusivity term cancels out with a stationary laser beam. Under this assumption, the problem is linearized to an inhomogeneous partial differential equation with constant coefficients: 14   [−𝛼𝐷 (𝜕2𝜕𝑥′2+𝜕2𝜕𝑦′2+𝜕2𝜕𝑧′2) +𝜕𝜕𝑡]𝜙(𝑥′, 𝑦′, 𝑧′, 𝑡)= 𝑞′′′(𝑥′, 𝑦′, 𝑧′, 𝑡)= 𝛼𝐷2𝑃0(1 − 𝛤)𝛼𝑧𝜋𝑘0𝑤𝑥𝑤𝑦𝑒−2𝛼𝑥′2𝑤𝑥2−2𝛽𝑦′2𝑤𝑦2𝑒−𝛼𝑧√𝛾𝑧′ (10) with  𝛼𝐷 =𝑘(𝑇)𝜌𝐶𝑝(𝑇) . (11)  Moreover, inclusion of a temperature dependent diffusivity in the model can be done through an additional time-variable transformation similar to the linearization of the 𝑘(𝑇) term in the Laplacian. Likewise, the true temporal profile of the temperature can be derived from the solution in linear-time domain to the as-linearized inhomogeneous differential equation. In this case, the temperature behaviour of thermal diffusivity couples the two temporal domains. When studying the temporal evolution of the temperature, the temperature dependence of thermal diffusivity should be treated explicitly following  𝕥 =1𝛼𝐷(𝑇0)∫𝛼𝐷 (𝑇(𝜙(𝒯)))𝜕𝒯𝑡0 (12) to linearize the temporal component in Eq. (10) to  1𝛼𝐷(𝑇)𝜕𝜙𝜕𝑡=1𝛼𝐷(𝑇0)𝜕𝜙𝜕𝕥 . (13)  The Green’s function of the standard diffusion equation for an infinite volume with Dirichlet boundaries as implemented and described by many [84]–[88], [94], [97] is 15   𝐺(𝑥′, 𝑦′, 𝑧′, 𝑡, 𝑥𝑠, 𝑦𝑠, 𝑧𝑠, 𝑡𝑠)= 𝐻(𝑡 − 𝑡𝑠) (14𝜋𝛼𝐷(𝑡 − 𝑡𝑠))32𝑒−(𝑥′−𝑥𝑠)24𝛼𝐷(𝑡−𝑡𝑠)−(𝑦′−𝑦𝑠)24𝛼𝐷(𝑡−𝑡𝑠)−(𝑧′−𝑧𝑠)24𝛼𝐷(𝑡−𝑡𝑠)    (14) , where 𝐻(𝑡 − 𝑡𝑠) is the Heaviside function. The authors of those works subsequently applied this formula to the semi-infinite problem with mixed boundary conditions (Neumann boundary at the surface where the optical input is imposed, i.e. thermally insulated surface, and Dirichlet at all others) by showing that a factor of two would enter the solution. Following the same approach, the complete linear temperature rise with the prescribed incident optical power is found to be  𝜙(𝑥′, 𝑦′, 𝑧′, 𝑡)= ∫𝜙0(𝑡, 𝑡𝑠) ( ∫ 𝜙𝑥𝜕𝑥𝑠∞−∞)( ∫ 𝜙𝑦∞−∞𝜕𝑦𝑠)( ∫ 𝜙𝑧∞−∞𝜕𝑧𝑠)𝜕𝑡𝑠𝑡−∞ 𝜙0(𝑡, 𝑡𝑠) = 2 (14𝜋(𝑡 − 𝑡𝑠))32 1√𝛼𝐷2𝑃0(1 − 𝛤)𝛼𝑧𝜋𝑘0𝑤02 𝜙𝑥(𝑥′, 𝑥𝑠, 𝑡, 𝑡𝑠) = 𝑒−(𝑥′−𝑥𝑠)24𝛼𝐷(𝑡−𝑡𝑠)𝑒−2𝛼𝜀𝑥𝑠2𝑤02 𝜙𝑦(𝑦′, 𝑦𝑠, 𝑡, 𝑡𝑠) = 𝑒−(𝑦′−𝑦𝑠)24𝛼𝐷(𝑡−𝑡𝑠)𝑒−2𝛽1𝜀𝑦𝑠2𝑤02 𝜙𝑧(𝑧′, 𝑧𝑠, 𝑡, 𝑡𝑠) = 𝑒−(𝑧′−𝑧𝑠)24𝛼𝐷(𝑡−𝑡𝑠)𝑒−𝛼𝑧√𝛾𝑧𝑠  (15) , where 𝑤0 is the normalized laser beam waist, and 𝜀 is the eccentricity of the laser, which measures the astigmatism of the elliptical Gaussian beam. The expressions for the two are  w0 = √wxwy (16) and  ε =wywx wx =w0√ε, wy = √εw0 . (17) 16   An expression can be derived for each spatial integral as  ∫ 𝜙𝑥𝜕𝑥𝑠∞−∞=√𝜋𝑡′√𝜀𝛼√2𝑡′𝑤02+1𝜀𝛼𝑒−12𝑡′𝑤02+1𝜀𝛼( √2𝑥′𝑤0)2 ∫ 𝜙𝑦𝜕𝑦𝑠∞−∞=√𝜋𝑡′√𝛽𝜀 √2𝑡′𝑤02+𝜀𝛽𝑒−12𝑡′𝑤02+𝜀𝛽( √2𝑦′𝑤0)2 ∫ 𝜙𝑧𝜕𝑦𝑠∞−∞= ∫ 𝜙𝑧𝜕𝑧𝑠∞0=√𝜋𝑡′2𝑒−𝛼𝑧2𝛾𝑡′4 𝑒−𝛼𝑧√𝛾𝑧′[𝑒𝑟𝑓𝑐 (𝛼𝑧√𝛾𝑡′2−𝑧′√𝑡′)] (18) , where 𝑒𝑟𝑓𝑐 (𝑥) is the complementary error function with 𝑡′ = 4𝛼𝐷(𝑡 − 𝑡𝑠) . The z-integral component obtained with exponential absorption is different from that in the literature following the same Green’s function approach. (Although such absorption behaviour has been applied by others [69]–[71], [81], [83], those works have been limited to cases of thermally-isotropic and temperature-independent media.) The temperature distribution along the z-axis (the optical axis into the medium) can have a similar Gaussian-like form as the x- and y-expressions if, instead of an exponentially decaying function, a Dirac delta function is used given which zero-light penetration into the medium is assumed. This abrupt absorption has been broadly assumed by many pursing the Green’s function method including Lu [84]–[87], Moody and Hendel [89], Lee and Albright [91], and Nissim et al. [88]. This results in a vastly simplified expression for the z-space integral as   ∫ ϕzδ ∂zs∞0= ∫ e−(z′−zs)24αD(t−ts)δ(√γzs) ∂zs∞0=12√γe−12w02t′( √2z′w0)2 . (19)  17  The complete linear temperature for the simplified abrupt absorption after rearranging the terms is then expressed as  ϕδ(x′, y′, z′, t) =2αDP0(1−Γ)π32√αβγk0∫e−12t′w02+1εα( √2x′w0)2−12t′w02+εβ( √2y′w0)2−12t′w02( √2z′w0)2√t′(t′+w02ε2α)(t′+εw022β)∂tst−∞ . (20)  Coordinate transformation of 𝜙𝛿(𝑥′, 𝑦′, 𝑧′, 𝑡) to 𝜙𝛿(𝑥𝑟 , 𝑦𝑟 , 𝑧𝑟 , 𝑡) gives a dimensionless spatial distribution scaled by the reduced normalized laser beam waist, as also done by Lu [84]–[87]:  𝜙𝛿(𝑥𝑟 , 𝑦𝑟 , 𝑧𝑟 , 𝑡)=2𝛼𝐷𝑃0(1 − 𝛤)𝜋32√𝛼𝛽𝛾𝑘0∫𝑒−𝑥𝑟2𝛼𝑡′𝑟02+1𝜀+−𝑦𝑟2𝛽𝑡′𝑟02+𝜀+−𝑧𝑟2𝛾𝑡′𝑟02√𝑡′ (𝑡′ +𝑟02𝜀𝛼) (𝑡′ + 𝜀𝑟02𝛽 )𝜕𝑡𝑠𝑡−∞ (21)  with   𝑟0 =𝑤0√2 (22) and  𝑥𝑟 =𝑥𝑟0= √𝛼 (𝑥′𝑟0) 𝑦𝑟 =𝑦𝑟0= √𝛽 (𝑦′𝑟0) zr =zr0= √γ(z′r0) .  (23)  18  Given that the starting point in time is −∞, the steady-state temperature distribution may be obtained by setting 𝑡 = 0 followed by a variable transform from 𝜕𝑡𝑠 to 𝜕𝑡′to yield  𝜙𝛿(𝑥𝑟 , 𝑦𝑟 , 𝑧𝑟) =𝑃0(1−𝛤)2𝜋32√𝛼𝛽𝛾𝑘0∫𝑒−𝑥𝑟2𝛼𝑡′𝑟02+1𝜀+−𝑦𝑟2𝛽𝑡′𝑟02+𝜀+−𝑧𝑟2𝛾𝑡′𝑟02√𝑡′(𝑡′+𝑟02𝜀𝛼)(𝑡′+𝜀𝑟02𝛽)𝜕𝑡′∞0 . (24)  An additional transform of the integral variable by 𝑢 =(αβ)14r0√t′ gives  ϕδ(xr, yr, zr) =P0r0⁄ (1−Γ)π32(αβγ2)14k0∫e−xr2√αβu2+1ε+−yr2√βαu2+ε+−zr2γ√αβu2√(u2+√βα1ε)(u2+√αβε)∂u∞0 . (25)  It is clear that the linear temperature rise is at its maximum at the origin, (𝑥𝑟 , 𝑦𝑟 , 𝑧𝑟) = (0,0,0), where the laser beam is impinged and decays off to zero at infinitely far away. In the application of thermionic emission, the linear surface temperature 𝜙𝛿(𝑥𝑟 , 𝑦𝑟 , 0) is the critical part of this expression as the non-linear emission current is substantially influenced by the temperature. Moreover, whether the loss mechanisms, namely, electron emission, conduction to the ambient and thermal radiation, are significant depends heavily on the temperature at the medium surface. Thus, the distribution of surface temperature and the peak temperature will be discussed in detail.    Analysis of the Significance of Absorption Profile To evaluate the impact of the different laser absorption profiles, the model considering abrupt absorption is juxtaposed to that with the exponential absorption function. The linear surface temperature with an exponentially decaying optical intensity into the medium is  𝜙(𝑥𝑟, 𝑦𝑟 , 0;  𝛼, 𝛽, 𝛾, 𝜀) =𝑃0(1 − 𝛤)𝑟0⁄𝜋32(𝛼𝛽𝛾2)14𝑘0∫ 𝑓(𝑢, 𝑥𝑟 , 𝑦𝑟 , 0)𝑔(𝑢, 0)𝜕𝑢∞0 (26)  with the definitions 19   𝑓(𝑢, 𝑥𝑟 , 𝑦𝑟 , 0) =𝑒−𝑥𝑟2√𝛼𝛽𝑢2+1𝜀+ −𝑦𝑟2√𝛽𝛼𝑢2+𝜀√(𝑢2 +√𝛽𝛼1𝜀) (𝑢2 +√𝛼𝛽 𝜀) (27) and  𝑔(𝑢, 0) = √𝜋(𝛿0𝑢)𝑒𝑟𝑓𝑐(𝛿0𝑢)𝑒(𝛿0𝑢)2 (28) with  𝛿0 =𝛼𝑧𝑟02(𝛾2𝛼𝛽)14 (29) , where 𝑓(𝑢, 𝑥𝑟 , 𝑦𝑟 , 0) is the integrand for the incomplete distribution integral which approximates the spatial variation of temperature on the medium surface using an abrupt absorption (as taken partially from the integrand of ϕδ(xr, yr, zr = 0) in Eq. (25) with 𝑧𝑟 = 0) and 𝑔(𝑢, 0) is the additional absorption correction function at 𝑧𝑟 = 0 stemming from the physical optical absorption, and δ0 is the coefficient for the absorption correction that determines how fast the correction function rises to unity. The complete distribution incorporates the correction function into the incomplete distribution integral and provides a more accurate solution to the actual physical problem when the parameters contributing to the coefficient of the correction function reside outside a given range. Numerical integration with 𝑔(𝑢, 0) is difficult as the exponent, e(u)2, accelerates rapidly beyond the numerical upper bound for double-precision floating-point data types. Thus, the asymptotic expansion of the complementary error function from Abramowitz and Stegun [98, p. 298] is used to evaluate the complete distribution integral following  erfc(x) ≅e−x2√π∑ (−1)m(2m−1)!!2m1x2m+1∞m=0 ≈e−x2√πx, x → ∞ . (30)  20  Therefore, in the integration algorithm, the absorption correction function is pinned to unity for any values of 𝑢 exceeding 21𝛿0. The integrands of the complete distribution function (𝑓(𝑢, 𝑥𝑟 , 𝑦𝑟 , 0)𝑔(𝑢, 0) and 𝑔(𝑢, 0)) are plotted in Figure 4 with varying coefficients of the absorption correction function.   Figure 4. Plot of 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0) (solid curves, left axis) and 𝑔(𝑢, 0) (dashed curves, right axis) evaluated at the origin (𝑥𝑟 = 𝑦𝑟 = 0). The thermal conductivity anisotropicity factors are 𝛼 = 𝛾 = 1 and 𝛽 = 100 with unity eccentricity, i.e. circular Gaussian beam.  It can be seen that the faster the correction function (dashed curves, Figure 4) ramps up to unity, the closer the abrupt-absorption solution to the physical reality. Thus, to be able to apply the simplified solution of a Dirac delta absorption profile, the laser penetration depth on the medium (i.e. inverse of optical absorption coefficient) must be relatively small compare to laser beam waist. The thermal anisotropicity of the medium also plays a factor, where a higher ratio along the optical path over the x- and y-axis on the surface is preferred, although to a lesser degree due to the power of 1/4. Two criteria, mainly associated with material properties of the medium, should be satisfied for a simplified abrupt-absorption approximation: 21   𝛼𝑧𝑟0 ≫ 2, ∴ 𝛿𝑧 ≪𝑟02 (31)   𝛾2 ≫ 𝛼𝛽 (32) , where 𝛿𝑧 is the penetration depth and should be much less than the laser beam radius at the medium surface. The effect of the correction coefficient on the complete distribution integral is delineated by the solid curves in Figure 4 where the area under the curve determines the value of the complete distribution integral, and the proportion of the area that each curve with different coefficients fills under the curve of 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0) indicates the accuracy of the simplification for dealing with absorption. The true temperature distribution can be deduced from the linear temperature solution through inverse Kirchhoff transform as defined previously. In the case of a constant, temperature-independent thermal conductivity, the true temperature is equivalent to the linear temperature with an offset of 𝑇0, the ambient temperature.   Analysis of Optical and Thermal Parameters It is clear from both physical intuition and the derived analytical solution that the thermal anisotropicity and the eccentricity of the Gaussian beam will broaden the surface temperature distribution in the respective direction. To investigate the effect of both exclusively on the peak temperature, the absorption correction function is assumed at unity regardless of the ratio between γ2 and αβ. Distortion to a circular incident Gaussian beam is implemented by tilting the angle of incidence around the y-axis. Increment in the tilting angle will stretch the laser beam waist along the x-axis. The change to normalized beam waist from the distortion is excluded from this analysis. The incomplete integral at the origin is formulated, by evaluating Eq. (27) at the origin, as 22   ∫ 𝑓(𝑢, 0, 0, 0)𝜕𝑢∞0= ∫ [(𝑢2 +√𝛽𝛼1𝑐𝑜𝑠 𝜃)(𝑢2 +√𝛼𝛽𝑐𝑜𝑠𝜃)]− 12𝜕𝑢∞0 (33) , where 𝜃 is the tilting angle between the medium surface and the optical wavefront for distortion and ranges from 0° (zero distortion) to < 90° (infinite beam waist along x-axis). Figure 5 (left) shows the integral evaluated with thermal anisotropicity varied from 1 to 3000 under different tilting angles. The maximum value for the incomplete distribution integral is calculated at unity anisotropicity among all three axes and eccentricity, and equals approximately 1.5708.  The distortion of the Gaussian beam plays a rather minor role as compared to the thermal anisotropicity. Only at extreme broadening of the beam waist does it lead to a relatively significant reduction in the outcome of the integral at the origin. On the other hand, the thermal anisotropicity casts comparatively larger influence on the distribution integral. From an isotropic medium to an anisotropicity of a single axis of 10000, the integral falls to below 0.4 from unity. The relationship between anisotropicity in logarithmic scale and the result of the integration closely follows a Gaussian shape. However, this does not include the product of the different anisotropicity factors outside the integral. Thus, an extended expression to Eq. (23) incorporating all the thermal anisotropicity terms outside the distribution integral described in Eq. (26) is presented in Figure 5 (right) described by   F(α, β, γ, θ) = (αβγ2)−14 ∫ [(u2 +√βα1cos θ) (u2 +∞0√αβcos θ)]− 12∂u . (34)  This addition elucidates the convoluted effect of thermal anisotropicity which exerts a profound influence on the temperature as compared to the tilting angle. For an one-dimensional medium, 23  such that 𝛼 = 𝛾, the effect of 𝛽 is even greater, given that the premise of abrupt absorption still holds.  Figure 5. Distribution integral at the origin, normalized with the scenario of unity eccentricity and anisotropicity (𝛼 = 𝛽 = 𝛾 = 𝑐𝑜𝑠 𝜃 = 1). Left: ∫ 𝑓(𝑢, 0, 0, 0)𝜕𝑢∞0 and right: 𝐹(𝛼, 𝛽, 𝛾, 𝜃). The thermal anisotropicity is swept by tuning 𝛽 while fixing 𝛼 and 𝛾 at unity. Note that the left and right panels are two independent figures.  Aside from the linear peak temperature, the as-derived analytical solution offers insights into the temperature distribution arising due to the induced laser beam. To study the effect of eccentricity and anisotropicity on the temperature distribution, the integral is numerically evaluated at discrete points in a manner similar to the above. The distribution as a function of dimensionless spatial coordinates in units of normalized beam waist is illustrated in Figure 6. The complete distribution integral is used considering an exponential absorption, although the influence of the absorption component is largely eliminated by tuning other parameters accordingly to maintain a fixed correction coefficient regardless of the amount of variation over either the anisotropicity or eccentricity. In essence, the product of 𝛼 and 𝛽 is fixed to 1 with varying 𝛼 and 𝛽 for each desired ratio; likewise, the normalized laser beam radius, 𝑤0, is kept constant while stretching 𝑤𝑦 and constricting 𝑤𝑥 for increasing eccentricity and vice versa.  24    Figure 6. Complete distribution integral over the reduced spatial coordinates. The distributions are plotted over xr and yr in units of 𝑤0 with 𝛽/𝛼 ratio (upper) and eccentricity (lower) swept from 1.5 to 20, as described by Eqs. (26) and (27), normalized by its peak value. Other parameters are 𝛼𝛽 = 1, 𝛾 = 1, 𝛿𝑧 = 11.1 𝜇𝑚 and 𝑤0 = 50 𝜇𝑚 (𝛿0 = 1.59).  25  Major distinctions between tuning the distribution through input distortion and thermal anisotropicity can be explicitly observed from Figure 6. In the latter case, the shape of the distribution appears to be preserved over varying ratios of anisotropicity. In other words, the distribution curves along both x and y axes do not cross over one another, and increasing or decreasing anisotropicity in the associated axis translates to a simple broadening or narrowing of the width of the profile over the entire spatial range. On the contrary, the distribution profile under the influence of the eccentricity of the incident laser beam shows a different behaviour as a result of the distortion in the corresponding axis. The distinction is most prominent at the higher ratio (cyan curve, Figure 6) where the profile along the x-axis has a reduced width near the laser beam center but is stretched farther away. A stronger coupling of the general gradient, or broadening, of the distribution function exerted by eccentricity is evident, although tuning the temperature distribution through optical means results in a more complicated response in the profile. Attempts to broaden the distribution via elongating the incident beam cross section will also result in a lower peak temperature, as shown previously (Figure 5), despite a fixed normalized beam waist. Another important motivation for deducing the temperature distribution is to affirm the impact of boundaries, having in mind practical applications where a finite geometry is involved. The distribution integral at the far end may not fall under a negligible level within a certain geometric limit as expected from the semi-infinite geometric assumption of this analytical model. Based on such assumption, the temperature rise will only diminish to zero at infinity; therefore, the application of this particular analytical model is valid for media where the geometric dimensions are large in relation to the reduced beam waist in the x and y direction, and the penetration depth along the z-axis. The amount of the residual temperature rise marks the error of the model as a result of the geometric assumption. The tolerance at the boundary along the axes on the surface plane is dependent on the associated anisotropicity and eccentricity, as detailed by   𝑥𝑟 → ∞, 𝜙(𝑥𝑟) ∝∫ 𝑒−𝑥𝑟2√𝛼𝛽𝑢2+1𝜀𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0)𝜕𝑢∞0∫ 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0)𝜕𝑢∞0≈ 0 (35)  26   𝑦𝑟 → ∞, 𝜙(𝑦𝑟) ∝∫ 𝑒−𝑦𝑟2√𝛽𝛼𝑢2+𝜀𝑓(𝑢,0,0,0)𝑔(𝑢,0)𝜕𝑢∞0∫ 𝑓(𝑢,0,0,0)𝑔(𝑢,0)𝜕𝑢∞0≈ 0 . (36)  The distance for which the linear temperature declines to below three orders of magnitude of that at the laser beam center on the surface can be within 500 laser beam radii on an isotropic medium (curve with 𝛼, 𝛽 = 1, Figure 7).    Figure 7. Normalized temperature profile (in relation to the peak point at 𝑥𝑟 = 𝑦𝑟 = 0) of complete distribution integral evaluated along the axis on the surface. When considering the effect of the absorption correction function, a penetration depth of 11.1 µm, a beam radius of 50 µm and unity 𝛾 are used. The actual values of the coefficient corresponding to each anisotropicity factor are listed in the legend.  Considering that the temperature of interest is on the order of thousands of Kelvins and a beam spot-size of 100 µm can be conveniently achieved with lasers and focusing lenses commonly available, a residual temperature of a few Kelvins at the boundaries of a 2.5 cm by 2.5 cm target area will result from an induced peak temperature of 3000 K at the beam center. Reducing the 27  anisotropicity in the respective axis increases the slope with which the temperature falls at increasing distance from the beam center, translating to a more localized temperature response.  The distribution along the z-axis is described by a more complicated form involving the error function:  𝑧𝛿 → ∞,𝜙(𝑧𝛿)∝∫ 𝑓(𝑢, 0, 0, 0) [√𝜋(𝛿0𝑢)𝑒𝑟𝑓𝑐 (𝛿0𝑢 −𝑧𝛿21𝛿0𝑢) 𝑒(𝛿0𝑢)2−𝑧𝛿] 𝜕𝑢∞0∫ 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0)𝜕𝑢∞0≈ 0 (37) with  𝑧𝛿 = 𝛼𝑧𝑧 =𝑧𝛿𝑧 (38) , where 𝑧𝛿 is the ratio of the location along z to the laser penetration depth. This formulation shows that the distribution is not determined by the size of the laser spot-size as in the two other axes, but rather the penetration depth. The distribution in this particular direction is greatly influenced by the absorption correction coefficient, determined by the ratios of laser beam radius to penetration depth and anisotropicity. To dissect the effect of each parameter, the temperature distribution in the medium along the optical axis is illustrated in Figure 8, Figure 9 and Figure 10 over varying penetration depth, laser beam waist, and the corresponding anisotropicity, 𝛾, respectively. The distribution given by the abrupt-absorption assumption follows a Gaussian-like integral form similar to that of the axes on the surface plane (e.g. Eq. (25)). By comparing the distribution curves against that of such assumption (dashed curve in the three figures), the validity of the simplified formulation by many [87]–[89], [91] in describing the temperature profile inside the medium can be attested. In the plots where the abrupt optical absorption is involved, the distribution is derived by scaling the z-coordinate in 𝜙𝛿(0, 0, 𝑧𝑟) from Eq. (25) by the 𝑟0𝛿𝑧 ratios associated to the values used in the exponentially decaying curves. 28  It can be explicitly shown that the actual temperature distribution inside the material is distinctively disparate from the one derived with abrupt absorption. With a large value of 𝛿0, through either minimizing the penetration depth and anisotropicity or widening the laser beam waist, a gradual trend for the temperature distribution to conform to the abrupt absorption case can be observed near the medium surface. The two eventually converge in terms of both the peak value and the distribution within a certain depth. These figures reveal the strong coupling between the boundary error along the z-axis and the parameters that determine the absorption correction coefficient, for which a lower value is preferred for minimizing the residual temperature at the depth limit. It is also found that the temperature at large distances from the surface is overestimated due to numerical error, i.e. the presented residual temperature at the finite boundary along the optical axis poses as the upper limit. 29   Figure 8. Absolute (upper) and normalized (lower) complete distribution integral evaluated along z-axis with unity eccentricity and anisotropicity in all three axes. The laser beam radius is fixed at 50 µm with the penetration depth swept from 0.02 µm to 500 µm.   30   Figure 9. Absolute (upper) and normalized (lower) complete distribution integral evaluated along z-axis with unity eccentricity and anisotropicity in all three axes. The penetration depth is fixed at 10 µm with the laser beam radius swept from 0.5 µm to 500 µm.   31   Figure 10. Absolute (upper) and normalized (lower) complete distribution integral evaluated along z-axis with unity eccentricity and thermal anisotropicity in the two peripheral axes while that of the respective z-axis is scaled from 0.01 to 1000. The laser beam radius and penetration depth are fixed at 50 µm and 10 µm, respectively.  32   Conversion to True Temperature The previous sections have studied the linear temperature, which is only a mathematical object derived from the Kirchhoff transform. In this section, the discussion will focus on the conversion to true temperature.  The temperature dependency of thermal conductivity couples the linear temperature to the true temperature. The temperatures of interest in thermionic emission reside in the range from room temperature (300 K) to 1700 K or even close to 3000 K, depending on the material’s electron emission behaviour and high-temperature stability. The thermal conductivity of many materials declines with increasing temperature due to increased scattering of heat carriers. In practice, the relationship can be highly non-linear and even non-monotonic.  Several works in the literature have studied the thermal conductivity of carbon nanotubes in both individual forms as well as ensembles. However, very few have experimentally explored the temperature dependency of thermal conductivity for aligned CNT forests at higher temperatures, i.e. above 1000 K, as illustrated in Figure 11. Most of these studies have focused on the low temperature (< 400 K) behaviour of thermal conductivity for CNT forests [99]–[102] and even fewer on the thermal anisotropicity [103]–[106] at either low or room temperature. On the other hand, Pop et al. have examined the thermal conductivity of individual SWCNTs from 100 K to 800 K through Joule heating [107]. Their analytical model manifests a steep thermal conductivity drop over temperature which was attributed to second-order three-phonon scattering in SWCNTs, not seen in non-metallic crystals which follow a more gradual, T-1, drop due to the predominant Umklapp scattering. The 𝑘(𝑇; 𝐿[𝜇𝑚]) relationship is described as [107]  𝑘(𝑇; 𝐿[𝜇𝑚]) = [(3.7 × 10−7)𝑇 + (9.7 × 10−10)𝑇2+9.3 (1 +0.5𝐿[𝜇𝑚])𝑇2]−1 (39) , where 𝐿 is the length of the nanotube in microns. 33   Figure 11. The temperature dependency of thermal conductivity of carbon nanotubes in the form of ensembles [99], [103], [105], [106], [108]–[110], bundles [100], [101], [111], [112], individual nanotubes [101], [107], [113]–[115], and sintered bulks [116]–[118] derived experimentally and theoretically [104], [119]–[121]. 34   Figure 12 illustrates the temperature behaviour with the tube length varied from 1 mm to 1 nm. The length-dependency of a SWCNT’s thermal conductivity is an interesting form for studying the effect of different types of thermal conductivity curve on the linear-to-true temperature relationship. By tuning the tube length, one can shift the location of the thermal conductivity peak from room temperature to as far as 791.9 K and 1435 K for lengths of 10 nm to 1 nm, respectively. This particular study will provide insights into the deviation of temperature distribution from the Gaussian-like distribution derived for a constant thermal conductivity model that gives a direct proportional conversion of linear temperature. Based on this high-temperature analytical formula for a SWCNT’s thermal conductivity, the linear-to-true temperature transformation can be formulated as  𝜙(𝑇) =1𝑘0∫𝑘(𝜏)𝜕𝜏𝑇𝑇0=1𝑘0∑𝑙𝑜𝑔 (𝑇 − 𝑤𝑇0 −𝑤)4𝑤𝑎2 + 3𝑎1𝑤: 𝑎2𝑤4+𝑎1𝑤3+𝑎0=0 (40) , where 𝑤 represents the roots of the specified quartic function. Since the Kirchhoff transform used in the linearization is normalized by the room-temperature thermal conductivity, presenting 𝑘0𝜙(𝑇) instead of just the linear temperature can be more convenient to capture the dynamics between the linear and true temperature regardless of the vastly different room-temperature thermal conductivity between different tube lengths. 35   Figure 12. The temperature dependency of thermal conductivity for an individual SWCNT with varying nanotube lengths. The term involving the nanotube length is associated to phonon-boundary scattering with a phonon mean free path of 0.5 µm [107].  Figure 13 portrays the relationship between 𝑘0𝜙(𝑇) and true temperature. The linear-to-true temperature transform of the temperature-dependent model is plotted against that of the linear model where a constant thermal conductivity is applied, adopted from the analytical expression, Eq. (39), evaluated at 300 K. In essence, the tangential slope of the transform, i.e.  ∆𝑇∆(𝑘0𝜙), is dictated by the thermal conductivity at the corresponding temperature of the point of tangency. A reduced conductivity value leads to a steeper rise in the linear-to-true temperature curve, which is physically intuitive, as low thermal conductivity entails restrictive heat conduction throughout the medium and inevitably results in an elevated temperature. This can be observed most prominently in curves corresponding to long nanotubes (> 0.5 µm) with a thermal transport behaviour strongly restrained by three-phonon-phonon scattering. 36   Figure 13. The relationship between linear temperature and true temperature with the analytical model from Ref. [107] for SWCNTs of different lengths, from room temperature to 6000 K. The solid curves represent the cases where a fixed thermal conductivity is used instead of the temperature-dependent model.   Another strikingly intriguing and significant insight to be drawn from the two 10-nm curves (blue, dashed and solid) in Figure 13 is the difference in the 𝑘0𝜙 values for the same corresponding true temperature. The product 𝑘0𝜙 is linearly proportional to the linear power density which can be defined as the total power divided by the reduced normalized Gaussian radius of the incident laser, i.e. 𝑘0𝜙~𝑃0𝑟0. Achieving 2000 K with a medium of constant thermal conductivity fixed at 183 W/mK (blue, solid) requires a linear power density of 310.9 kW/m whereas a minimum linear power density of twice that value (~697 kW/m) is necessary to obtain the same temperature on the temperature-dependent medium (blue, dashed). However, this dichotomy in the laser heating efficiency converges at around 5725 K, and beyond this point, the medium with temperature-dependent thermal conductivity becomes more efficient for laser-induced heating. For instance, the linear optical power density to achieve 6000 K is 1042 kW/m for the constant-conductivity case, slightly above the 999 kW/m requirement for the temperature-dependent-conductivity case. This trend implies that despite having a higher thermal conductivity at room temperature, if a material has a severely reduced thermal conductivity with 37  increasing temperature, ultimately, a higher temperature can be achieved with less linear power density as compared to a material with a lower fixed thermal conductivity at room temperature. A more dynamic temperature dependency, such as that for 10-nm SWCNTs, results in a more complex thermal response at high linear power density. Even though the thermal conductivities for all tube lengths converge at 3000 K and beyond (see Figure 12), a conspicuous disparity in the linear power density required for attaining a temperature of 3000 K to 6000 K is evident. Thus, it is vital to examine not only the thermal conductivity at the target temperature, but also the comprehensive temperature dependency encompassing the entire range starting from the ambient temperature.   Loss Correction for the Laser-Induced Heating Model The analytical model derived in sections 2.1 and 2.2 presents the solution to the non-linear inhomogeneous heat equation with a Gaussian heat generation term. It is imperative to note that the model does not include loss mechanisms of any kind at the boundary. This is primarily valid either at very high input optical intensity such as in laser machining where thermal loss is comparably negligible, or in low-temperature applications. However, this is not necessarily true in the case where even a low-power handheld laser could induce a sizable temperature [49]. This leads to a situation where the loss, particularly radiative loss, plays a critical role in the thermal analysis of the problem. Three common heat loss mechanisms are convection, conduction and radiation. The former two are confined to conditions where the Knudsen number, the ratio of the molecules’ mean free path and the characteristic dimension of the system, is small. The ambient condition for most applications involving charged particles is limited to high vacuum, a pressure below 10-6 Torr, in order to prevent arcing and electron beam-spread by the adverse beam-gas interaction. Thermal radiation, on the other hand, cannot be avoided by a reduced ambient pressure. The radiative heat flux is expressed by the Stefan-Boltzmann law,  𝑞𝑟𝑎𝑑′′ = 𝜖𝑡𝜎(𝑇4 − 𝑇04) (41) , where 𝜖𝑡 and 𝜎 are the emissivity of the radiating surface and the Stefan-Boltzmann constant, respectively. This quartic relationship implies an imperceptible energy loss at low temperatures, and incorporating this effect is only necessary in surfaces reaching high temperatures. A 38  common approach is to incorporate the radiation only at the medium-vacuum boundary. However, to avoid the non-linear boundary condition arising from 𝑇4 and 𝑘𝑧(𝑇) terms, a simpler method is to embody the radiative loss through an additional step after calculating the peak temperature from a set of given parameters. The temperature profile induced by a given input laser beam can be solved for by a self-consistent method with a sequence of calculations based on the following:  𝑃0𝑖 = 𝑃0, 𝑖 = 0  𝜙𝑚𝑎𝑥(𝑃0𝑖; 𝑘0, 𝛤, 𝑟0, 𝛼, 𝛽, 𝛾, 𝜀)=𝑃0𝑖(1 − 𝛤)𝑟0⁄𝜋32(𝛼𝛽𝛾2)14𝑘0∫ 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0)𝜕𝑢∞0  𝜙(𝑥, 𝑦) =𝜙𝑚𝑎𝑥 ∫ 𝑓(𝑢, 𝑥, 𝑦, 0)𝑔(𝑢, 0)𝜕𝑢∞0∫ 𝑓(𝑢, 0, 0, 0)𝑔(𝑢, 0)𝜕𝑢∞0  𝑇(𝑥, 𝑦)𝑇⟻𝜙↔  𝜙(𝑥, 𝑦)  𝑇(𝑥, 𝑦) = (𝑇𝑚𝑎𝑥 − 𝑇0)∆𝑇(𝑥, 𝑦)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ + 𝑇0  (42) 39  𝑃𝑟𝑎𝑑 = 𝜖𝑡𝜎∬(𝑇(𝑥, 𝑦)4 − 𝑇04)𝜕𝑥𝜕𝑦= 𝜖𝑡𝜎 ((𝑇𝑚𝑎𝑥 − 𝑇0)4∬∆𝑇(𝑥, 𝑦)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅4𝜕𝑥𝜕𝑦+ 4(𝑇𝑚𝑎𝑥 − 𝑇0)3𝑇0∬∆𝑇(𝑥, 𝑦)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅3𝜕𝑥𝜕𝑦+ 6(𝑇𝑚𝑎𝑥 − 𝑇0)2𝑇02∬∆𝑇(𝑥, 𝑦)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅2𝜕𝑥𝜕𝑦+ 4(𝑇𝑚𝑎𝑥 − 𝑇0)𝑇03∬∆𝑇(𝑥, 𝑦)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅𝜕𝑥𝜕𝑦)  𝑖 = 𝑖 + 1  𝑃0𝑖 = 𝑃0𝑖−1 − 𝑃𝑟𝑎𝑑  , where 𝑇𝑚𝑎𝑥 is the peak temperature at the center of the laser input, ∆𝑇(𝑥, 𝑦)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ is the normalized distribution of temperature rise, and 𝑃𝑟𝑎𝑑 is the total radiative power loss. In essence, the ultimate peak temperature is constrained by thermal radiation at elevated temperatures. This constraint, however, is rather imperceptible in materials with generally high thermal conductivity. A high conductivity leads to a reduced linear power density-to-temperature conversion and accordingly allows a greater power margin to account for radiative loss.   Associated Error for Radiative Loss Even with the inclusion of a distributive radiation loss, a major underlying origin of error is ascribed to the premise of how the energy source term is defined in the inhomogeneous equation. The Gaussian input is assumed to be intact regardless of the distribution and intensity of the incurred thermal radiation. However, this is a critical assumption to simplify the non-linearity of the problem to account for the radiation loss on top of the analytical framework. The root-mean-squared error over the region discretized with a 5-by-5 µm2 grid over a 1-by-1 mm2 area is 40  shown in Figure 14. The area is sufficient for the calculation of radiation which is only a concern at very high temperatures, that is, in the region near the center of the laser spot.  Figure 14. The peak temperature versus input optical power and the associated root-mean-squared error for different combination of thermal conductivities.  Increasing the anisotropicity leads to a more eccentric thermal profile that translates to a thermal radiation distribution much distorted from the Gaussian input. Ultimately, this will exacerbate the associated error at high optical power. Another inherent issue for such treatment of thermal radiation is the diverting shape of temperature profile along the optical axis. The exponential decaying intensity profile is preserved as an underlying assumption and is scaled proportionally to the reduced peak intensity map at the surface. This assumption is legitimate in the particular case involving an abrupt absorption. Other errors come from presumed linearity over a few physically non-linear material constants, such as the absorption coefficient, surface reflectivity and emissivity. The absorption coefficient and reflectivity are both a function of temperature in practice but only constants are used throughout the model. The emissivity also depends on temperature although a constant emissivity evaluated at a particular temperature is used in the model. The analytical model also relies on a fixed ratio of thermal anisotropicities, meaning that the thermal conductivities along the different axes follow the same dependency on temperature. 41  To verify the analytical model and the correction for radiation, a comparative finite-element analysis (FEA) is performed using the heat-transfer module in COMSOL Multiphysics® (Figure 15) [122]. A three-dimensional medium is set-up in COMSOL with Dirichlet boundary conditions at the medium surfaces except the one where the Gaussian heat source is applied to account for thermal radiation. The resolution of the model is dictated by computational power which, in turn, limits the spot-size of the laser beam in proportion to the geometric size of the medium. To mitigate the computational intensity without greatly compromising the accuracy of the solution, a gradient meshing around the laser spot is used.   Figure 15. Comparison between analytical solutions to that of FEA at different thermal conductivities. A medium with area of 20 mm by 10 mm and a 20-mm thickness is used as the FEA geometry and the numerical integration area of radiative loss in the analytical model. An absorption coefficient of 0.09 µm-1 and unity emissivity with no reflection loss are used.    42       Chapter 3   Application of the Analytical Approach to Different Materials    Bulk Specimen: Isotropic (Metal/Semiconductor) and Planar (Pyrolytic Graphite) In this section, the as-refined analytical model is applied to compute the peak temperature over a range of input laser powers in a variety of materials. Known for their exceptional high-temperature stability, refractory metals are good candidates as high temperature laser-heating targets. Thermal conductivities for these metals vary very slowly with temperature above room temperature, making them good candidates for studying models with moderate temperature dependency. Monocrystalline silicon and germanium, although having a much lower melting point, represent a thermal conductivity following precisely a steep, T-1 trend. The thermal conductivities of these materials are illustrated in Figure 16, revealing drastic differences in the behaviour of thermal conductivity for different materials within the temperature range of interest. The other material properties are shown in Table 2. Emissivity is only included in the calculation of radiative loss, i.e. we will use the effective absorbed optical power as the input power and assume zero reflection at the surface of the medium. 43  Table 2. Material properties of the listed materials. The quoted emissivity values in the table are designated to total hemispherical emissivity with the exception of germanium where only spectral emissivity at 0.65 µm and that of molten specimen are available. The absorption coefficients are derived from the extinction coefficient based on a 532-nm source.  Tungsten Molybdenum Niobium Silicon Germanium 𝑘0 (W/mK) [123] 174 138 53.7 148 59.9 Tmelting (K) 3660 2800 2750 1685 1200 Total Emissivity 0.241-0.282 (2000-2600 K) [124] 0.217-0.274 (2000-2600 K) [124] 0.225-0.299 (2000-2600 K) [124] 0.73 (1475 K, 600 Ωcm) [125] 0.17 (1211.3 K, molten Ge) [126],  0.53 (1200 K, 0.65 µm) [127] Absorption Coe. (107 m-1, 532 nm) 6.4249  [128, p. 366]  8.6539  [128, p. 312] 6.7984  [129, pp. 407–408] 0.0765 (i) [130], 0.1028 (26 Ωcm) [131, pp. 564–565]  5.6868 (i) [132, pp. 473–474]  Graphite is also a very intriguing material candidate to study due to a number of factors. First, the material exhibits a constant thermal anisotropicity of approximately 3x10-3 with respect to the basal plane, as realized by fitting the thermal conductivity data of the two directions reported by Ho et al. [123]. The exceptional temperature stability of graphite is well recognized with its melting point commonly reported as being around 5000 K [133]. A few works have attempted to deduce the total emissivity of graphite. Jain and Krishnan have measured both the spectral and total emissivity (~0.867 at 0.65 µm and 0.833) of Acheson graphite up to 2100 K by collecting the radiated intensity of a heated graphite tube at the surface and from a cavity with an optical pyrometer [134]. The hemispherical spectral emissivity of pyrolytic graphite at 0.65 µm is reportedly in the range of 0.6 to 1, depending on the surface condition after exposure to argon at elevated temperatures [135]. Variations such as surface roughness also affect the emissivity of the specific graphite specimen. For instance, polished and sandblasted graphite have emissivities of 0.75 and 0.85, respectively at 1800 C [136]. Herein, an emissivity value of 0.80 is chosen for the analysis of graphite. An extinction coefficient between 1.3 to 1.4 over the visible range for the basal plane of pyrolytic graphite can be found across different studies in the literature [137], [138]. This translates to an absorption coefficient of 3.157x107 m-1 for a 532-nm incident wavelength. 44   Figure 16. Temperature dependent thermal conductivity for different materials. The data were taken from Ref. [123].   The peak surface temperature in these media as a function of optical power is depicted in Figure 17, along with the ratio of the radiative loss to the input power. The peak-surface-temperature figures are derived from complete absorption at the surface with an incident beam radius of 50 µm. It can be observed that the materials capable of reaching a high temperature at low optical input tend to be non-metals, although silicon and germanium are not ideal for high-temperature applications. Pyrolytic graphite, on the other hand, is capable of withstanding a temperature of thousands of Kelvins, depending on the environmental conditions and impurities. However, several Watts of optical power are required even for graphite. The radiative loss ratio of the different materials highlights the significance of incorporating thermal radiation in the analytical model where efficient laser-induced heating is involved. 45   Figure 17. Peak surface temperature versus input optical power for the listed materials. The radius of the illuminated spot was … in all cases. The left vertical axis applies to the solid curves and the right vertical axis applies to the dashed curves.    Carbon Allotropes – Graphitized and Amorphous Comparing the thermal response of different allotropes of carbon can provide guidance as to the influence of dimensionality on laser-induced heating. The thermal conductivity of individual SWCNTs as given by Pop et al. [107] is used for modelling a CNT forest after scaling with a volumetric fill factor of 9%. The influence of tube length on the peak induced temperature is illustrated in Figure 18. As we can see from Figure 12 Figure, shorter tubes lead to lower conductivity in the temperature range of interest. In the case of a nanotube forest with macroscopic heights, the usage of the thermal conductivity of shorter tubes implies that each long nanotube is composed of many short segments, leading to higher levels of phonon boundary scattering between adjacent segments and overall lower thermal conductivity. The other material constants are extracted from the literature as described in the following. The absorption coefficient of a CNT-based ensemble is determined by its density, as evident from the work of Jeong et al. with SWCNT solutions [139]. In equivalent terms, the fill factor of 46  nanotubes is a key factor in determining the absorption coefficient of an aligned array of CNT (i.e. CNT forest). Sample-to-sample variation in the fill factor can easily translate into vastly different absorption properties of the forest. Even with the controllability offered by thermal chemical vapour deposition, statistical variation in the absorption coefficient of the as-grown CNT forest can be greater than 3-fold among 11 synthesis runs following the same growth recipe [140]. Although Park et al. were able to deduce the comparative changes in the density using capillary densification, the absolute fill factor of the CNT forest was not reported in relation to the absorption coefficient. It is inherently difficult to estimate the exact absorption coefficient at a specific point on a specific sample that is composed of smaller structures with a length scale of microns to millimeters in length and nanometers in width. To limit the number of unknowns, an absorption coefficient of 0.09 µm-1 from the pre-densified sample in Ref. [140] is assumed for all the nanotube-based media in this work. Unity emissivity [141] is also assumed for CNT forests.  Figure 18. Peak surface temperature versus input optical power and the associated radiative loss using the thermal conductivity of individual SWCNTs based on Pop’s formula [107] with a fill factor of 9% applied to all cases. The radius of the illuminated spot was 50 µm in all cases. The left vertical axis applies to the solid curves and the right vertical axis applies to the dashed curves.  47  Unsurprisingly, optical constants for amorphous carbon have an even greater variation depending on the specific specimen. Sails et al. noted an experimental absorption depth of 204 nm (~4.9x106 m-1) with an Ar+ laser at the 514.5-nm line [142] and the value is reportedly affected by the grain size [143]. To be consistent with the graphite sample, the absorption coefficient of amorphous carbon was chosen from Stagg and Charalampopoulos’ [138] (2.3176x107 m-1 at 532 nm). Unity emissivity is also assumed for amorphous carbon. Figure 19 shows the peak surface temperature under varying optical powers for different allotropes of carbon. The curve for amorphous carbon is limited at 1500 K as it is the onset temperature of graphitization [123, p. 110] and its thermal conductivity, in practice, will gradually rise at higher temperature until it conforms to that of graphite. A distinguishable difference among three allotropes can be observed with each type reaching a peak temperature of thousands of Kelvins at a different optical power level with amorphous and graphitic carbon being the most and least efficient targets for laser-induced heating.   Figure 19. Peak surface temperature versus input optical power for different types of carbon-based targets, include one-dimensionally anisotropic, isotropic and planar. The radius of the illuminated spot was  in all cases.  48   Effect of Thermal Conductivity on Temperature To isolate the effect of thermal conductivity, the 1-mm curve is taken from Figure 18 and compared with the hypothetical scenarios of a temperature-invariant thermal conductivity taken from the midpoint of the thermal conductivity values of the 1-mm curve between 300 to 5000 K, and a temperature dependency inverted around this midpoint (Figure 20). It has to be noted that the temperature at which the three scenarios intersect over the input laser power does not necessarily translate to the temperature where the thermal conductivities of the three cases coincide. This is because the linear temperature follows the anti-derivative of the thermal conductivity’s temperature dependency rather than a having a direct relation.    Figure 20. Peak surface temperature versus input optical power for different temperature dependencies of thermal conductivity. The thermal conductivities of these different scenarios are presented in the inset. The radius of the illuminated spot was 50 µm in all cases.  A point of great interest is the effect of the exact nature of decay of thermal conductivity with temperature, on heating. In particular, we know that two-phonon Umklapp phonon scattering leads to a T-1 behaviour as commonly seen in bulk materials, whereas three-phonon scattering 49  can lead to a T-2 behaviour as seen in carbon nanotubes. To investigate this, a further study with artificial conductivity functions is carried out with the test functions  𝑘0 = 1 [𝑊/𝑚𝑘] (43)    𝑘𝑇−1 =𝑘00.001(𝑇 − 300) + 1[𝑊/𝑚𝑘] (44)    𝑘𝑇−2 =𝑘00.001(𝑇 − 300)2 + 1[𝑊/𝑚𝑘] (45) , as plotted in Figure 21 which shows the decay rates in thermal conductivity of each artificial function. The temperature distribution is calculated for each of these cases under different levels of anisotropicity. The objective here is to emulate the reduction in the dimensionality of the target medium by elevating the thermal anisotropicity accordingly. The outcome for the case of constant thermal conductivity is illustrated in Figure 22, while the cases presented in Figure 23 and Figure 24 demonstrate the change in the temperature profile with first-order and second-order fall of thermal conductivity, when the material's dimensionality is reduced from 3 to 2 to 1.  50   Figure 21. The three artificial thermal conductivity functions of Eqs. (43), (44) and (45) from 300 K to 5000 K.  The different combinations of thermal anisotropicity in Figure 22, Figure 23 and Figure 24 thus correspond to different dimensionalities of the material, i.e. the blue curve represents a one-dimensional material such as a carbon nanotube forest and the red curve represents an isotropic bulk. To consider the two-dimensionally anisotropic scenario, both 𝛾 and 𝛽 are modified separately, as portrayed by the green and purple curves. The thermal anisotropicity coefficient along the x-axis, 𝛼, is fixed to unity in all cases. It is observed that the overall temperature gradient along the x-axis is increased as a result of limiting thermal conduction along the other axes. The most significant interpretation from these different scenarios is that a high gradient in temperature in a certain direction can be achieved without a reduction in the thermal conductivity corresponding to that particular axis; by simply limiting the thermal conductivity in the “off-axis” directions, an elevated efficiency in heating via laser can be achieved. This is a key observation and serves as the basis for an entirely new approach to maintaining a significant temperature gradient in thermionic and thermoelectric (and potentially other phononic) devices. The uniqueness of this approach is in that the thermal conductivity in the axis of interest remains unchanged. Normally, an attempt to create a thermal gradient by reducing the thermal 51  conductivity in the direction of interest will also lead to a drop in the electrical conductivity, which is detrimental in energy conversion applications. This effect has, indeed, been the main challenge in thermoelectric conversion over the past few decades and it is this very effect that approaches such as creating EC-PG structures attempt to address. Here, it is seen that a high temperature gradient can be attained through changing the dimensionality of the medium (for example by micro/nano patterning) or using materials with inherently low dimensionality such as carbon nanotube forests. Furthermore, by comparing Figure 22, Figure 23 and Figure 24, it is observed that a decrease in thermal conductivity with temperature leads to more efficient localized heating, with a higher peak temperature and temperature gradient; a steeper decrease in thermal conductivity leads to an even higher peak temperature and temperature gradient. Moreover, it is seen that this effect is severely exacerbated in low-dimensional materials: in Figure 24, the input optical power is more than an order of magnitude lower than in Figure 22 and Figure 23, leading to negligible heating in the three-dimensional and two-dimensional cases, but a much higher peak temperature and temperature gradient in the one-dimensional case compared to Figure 22 and Figure 23. This further supports the mechanism originally proposed for the Heat Trap effect based on a positive feedback mechanism arising from low-dimensionality and a fast drop of thermal conductivity with temperature [11], [49].  52   Figure 22. Temperature distribution along x-axis (upper) and y-axis (lower) for different thermal anisotropicities (corresponding to different material dimensionalities) in the case of constant thermal conductivity. The radius of the illuminated spot was 50 µm in all cases.  53   Figure 23. Temperature distribution along x-axis (upper) and y-axis (lower) for different thermal anisotropicities (corresponding to different material dimensionalities) in the case of a first-order decreasing thermal conductivity. The radius of the illuminated spot was 50 µm in all cases.   54   Figure 24. Temperature distribution along x-axis (upper) and y-axis (lower) for different thermal anisotropicities (corresponding to different material dimensionalities) in the case of a second-order decreasing thermal conductivity. The radius of the illuminated spot was 50 µm in all cases.   Choice of Materials for Applications in Thermionic Cathodes  In applications specific to thermionic emission where the temperature of the emitting cathode must be elevated to several thousands of Kelvins, materials with high optical power-to-55  temperature conversion efficiency are needed. Thus, attributes reflecting this key criterion must be investigated for comparing cathode materials. Such a measure can be formulated based on the materials’ thermal conductivity behaviour alone as  𝐿𝑃𝐸𝑘(𝑇) =∆𝑇𝑘0𝜙(𝑇)=𝑇 − 𝑇0∫ 𝑘(𝜏) 𝜕𝜏𝑇𝑇0 (46) , and  𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑡𝑖𝑣𝑒 𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑣𝑒 𝐿𝑜𝑠𝑠 = ∫ ?̃?−1(𝜃) 𝑑𝜃𝜙0−∫𝑇 − 𝑇0𝜙𝜃 + 𝑇0𝑑𝜃𝜙0=1𝑘0[∫ 𝜏𝑘(𝜏) 𝑑𝜏𝑇𝑚𝑇0−𝑇0 + 𝑇𝑚2∫ 𝑘(𝜏)𝑑𝜏𝑇𝑚𝑇0] (47) , where ?̃?−1(𝜃) denotes the inverse Kirchhoff transform described by the temperature dependency of the material. The first criterion, 𝐿𝑃𝐸𝑘(𝑇), elucidates the rise of temperature in Kelvin at the specimen surface given a linear power density in W/m excluding other linear constants, which will be discussed in the following. The second equation provides a non-physical measure of the desirability of the material with respect to the reduction of radiation over the temperature distribution as a direct result of the material’s temperature dependency of thermal conductivity. It is computed by the integral difference between the linear-to-true temperature conversion from the non-linear temperature dependency and that of a linear thermal behaviour. Materials with a lower value of such measure are preferential for thermionic applications. A positive loss indicates that the temperature will be more diffused than if the material is of a constant thermal conductivity with the same linear power efficiency and, since the thermionic electrons are more susceptible to temperature than radiative photons through incandescence, any positive loss implies an undesirable material choice. It is with certain generalization that the materials with a decreasing thermal conductivity beyond room temperature appear to be a better choice over those with increasing thermal conductivity. The required optical power, however, 56  also poses a major constraint on the material choice as a light-induced thermionic source, as can be observed among the disparate values between different groups of materials listed in Figure 25.   Figure 25. Linear power efficiency and the corresponding comparative radiative loss for different materials over varying temperatures. The temperature, starting at roughly 300 K (as indicated by the vertical dashed line at zero radiative loss), is varied from 500 K to 3000 K in steps of 500 K. For materials with no conductivity data available at or beyond the swept temperature points, the highest temperature is used as the last point. Inset: the temperature-dependent thermal conductivity of the materials studied herein. For anisotropic materials (marked by *), the thermal conductivity is re-scaled accordingly to yield 𝛼𝛽𝛾2 = 1. The single-axis bar-plot presents a collage of the ranges of linear-power-density to surface peak temperature rise corresponding to each material over the temperature range of interest.  On the other hand, materials with low-loss thermal conductivity behaviour (steep decline in the thermal conductivity with increasing temperature) may not possess a high linear power efficiency and would require a high-intensity light source to reach a high temperature. Thus, both of these attributes are vital in selecting the best material for specific applications. For instance, with an exceptional linear power efficiency, orders of magnitude greater than that of refractory metals and semi-metallic crystals, amorphous carbon demonstrates good characteristics and serves as a great candidate for light-induced heating applications where high temperature can be 57  conveniently attained by a rather low-powered light source, despite the fact that it may not be preferred for a minimum temperature spread given its escalating dependency on temperature. In addition, the practical usage of amorphous carbon is constrained by its tendency to graphitize at elevated temperatures. Thus, when assessing different material candidates for applications involving light-induced heating, one has to be aware of the different attributes of the thermal conductivity of the material of choice.  The thermal conductivity of MWCNT forests is taken from the room-temperature laser-flash measurement carried out by Ivanov et al. [102] along both longitudinal and transverse directions. Most others in the literature reporting the experimental values of the CNT forest’s thermal conductivity have generally focused on the longitudinal conductivity, although Jakubinek et al. did report one transverse measurement [103] carried out on a different CNT forest from those used in the longitudinal measurements. Thus, Ivanov’s value is arguably as the most relevant one found in the literature for experimentally derived anisotropic thermal conductivity of CNT forests. With a constrained thermal conductivity in the transverse direction, the MWCNT forest attains a linear power efficiency that is orders of magnitude above that of all other materials listed in this work. Beyond the influence of non-linear thermal conductivity, the other components of the analytical model at extreme ranges can have a pronounced effect on the resulting linear temperature as previously described. The convoluted effect of other primary linear parameters, including the penetration depth and thermal anisotropicity scaled over different axes, is summarized using the following formulation  LPElinear = k0ϕP0w0⁄=∫(w04πδz√αβu)erfc(√γ2√2δzw0(αβ)14u)(  eγ8(δzw0)2√αβu2)  √(u2+√βα)(u2+√αβ)∂u∞0 . (48)  58  As the outcome of such a product is linearly proportional to the effective linear power density, this suggests that a higher value of 𝑘0𝜙𝑃0𝑤0⁄ is preferential for thermionic applications. It can be explicitly observed that the product peaks around the centerline where the anisotropicity is reduced to unity at lower ratios of penetration depth to laser spot-size. Considering a 2D material like graphite where the C-axis is parallel to the optical axis of a normal-incidence source, the thermal anisotropicity coefficients on the x-y plane (𝛼, 𝛽) would be greater than 1 while the anisotropicity along the z-axis is held at unity. Conversely, if the medium is orientated in a way that the optical axis is parallel to the basal plane, then 𝛾 and either 𝛼 or 𝛽 would be greater than 1. These two scenarios are marked by the vertical solid white lines in Figure 26 from which a discernible difference can be observed at especially minimal 𝛿𝑧. The same phenomenon is similarly witnessed for a 1D material where the non-unity thermal anisotropicity is either 𝛽 (optical axis parallel to one of the transverse directions of the material) or 𝛾 (optical axis parallel to the longitudinal direction of the material), as shown by the vertical dashed line. However, as the peak temperature may be located inside the medium, it is also crucial to approach the same study over the peak value instead of evaluating that at the surface. Figure 27 illustrates the linear parameters’ influence on the peak values of 𝑘0𝜙𝑃0𝑤0⁄ and its associated location inside the medium. This observation of the influence of material orientation is particularly crucial when examining different experimental configurations involving anisotropic materials. 59   Figure 26. Effect of the other parameters on the effective linear power density at the surface of the medium with unity eccentricity. Each quadrant denotes a different anisotropicity scaling approach. Only one or two anisotropicity coefficients are tuned while fixing the other(s) in each quadrant.  60   Figure 27. Effect of the other parameters on the effective linear power density at the depth at which the peak values are located. The penetration depth is with respect to the laser beam waist while the depth, ZZmin, is normalized to the penetration depth.     61       Chapter 4   Conclusion, Summary, and Future Work   Laser-induced heating poses as an interesting phenomenon with direct application in many areas of technology. By investigating the thermal behaviour of a variety of materials beyond traditional bulk metals or semi-metallic crystals that have been perceived as the default cathode material in machining, optics and semiconductor industries, this work investigates laser-induced heating and sheds light on how to approach developing an efficient photo-thermocathode. The derived analytical model allows one to consider temperature dependency, thermal anisotropicity, eccentricity, and absorption depth. The relationships between these parameters and the peak temperature as well as the temperature distribution are elucidated in this study. Thermal radiation is also considered through an additional step to yield a non-linear relation of temperature against the input power that is characteristic in efficient optical-heating targets. Peak temperatures for different types of materials with varying dimensionality and temperature-dependency have been presented using the derived model and it was seen that reducing the dimensionality can have a drastic effect on heating. An overall low thermal conductivity obviously helps one achieve high temperature, but a diminishing trend over temperature reduces the associated radiative loss at high temperatures and can lead to substantially higher peak temperatures and temperature gradients. Additionally, the orientation of an anisotropic medium with respect to the optical axis of the input laser can have an influence on the resulting temperature profile. These findings have 62  important, broad implications in applications concerned with heating, including thermionic and thermoelectric energy conversion and phononics. This analytical approach can be expanded to finite geometries to take into account the boundary effects. To eliminate the errors arising from assuming a fixed Gaussian profile for the input power, one must include thermal radiation as the proper boundary condition at the interface. Following such approaches requires solving the non-linear inhomogeneous boundary problem, a result of the temperature-dependent thermal conductivity along the z-axis and the quadratic Stefan-Boltzmann law. A major limitation of this analytical model lies in the assumption that the thermal anisotropicity of the medium is independent of temperature, as analytical solutions without such assumption are challenging to derive. Other non-linearities in the optical parameters such as the absorption coefficient and emissivity can be introduced to attain a more physically realistic model. Additionally, the inclusion of thermionic energy losses will be essential for simulating the cathode temperature accurately under biased conditions. These further developments can help build an analytical thermal model to fully study photo-thermocathodes in a computationally effective, versatile and accurate manner. Subsequent experimental work on VA-MWCNT forests and other low dimensional and non-linear materials can be used to validate the findings of this theoretical analysis.      63    References  [1] H.-J. Schwarzmaier, F. Eickmeyer, V. U. Fiedler, and F. Ulrich, “Basic Principles of Laser Induced Interstitial Thermotherapy in Brain Tumors,” Med. Laser Appl., vol. 17, no. 2, pp. 147–158, 2002. [2] A. M. Mols, V. Knappe, H. J. Buhr, and J.-P. Ritz, “Laser-induced Thermotherapy (LITT): Dose-Effect Relation on Lung Tissue,” Med. Laser Appl., vol. 19, no. 3, pp. 160–166, 2004. [3] A. Fasano, D. Hömberg, and D. Naumov, “On a mathematical model for laser-induced thermotherapy,” Appl. Math. Model., vol. 34, no. 12, pp. 3831–3840, Dec. 2010. [4] J. Meijer, “Laser beam machining (LBM), state of the art and new opportunities,” J. 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