A three-level finite difference scheme for solving a dual-phase-lagging heat transport equation in spherical coordinates

Heat transport through thin films or micro-objects is of vital importance in microtechnology applications. For instance, metal thin films are important components of microelectronic devices. The reduction of the device size to microscale has the advantage of enhancing the switching speed of the device. On the other hand, size reduction increases the rate of heat generation, which leads to a high thermal load on the microelectronic devices. Heat transfer at the microscale is also important for the thermal possessing of materials with a pulsed-laser. Examples in metal processing are laser micromachining, laser patterning, laser processing of diamond films from carbon ion implanted copper substrates, and laser surface hardening. In thermal processing of materials, microvoids may be found owing to thermal expansion. When such defects begin in the workpiece, their thermal energy in the neighborhood of the defects may be amplified, resulting in severe material damage and, consequently, total failure of the thermal processing. A detailed understanding of the way in which the local defects dissipate the thermal energy is then necessary not only to avoid the damage but also to improve the efficiency of the thermal processing. The heat transport equation at the microscale is different from the traditional heat diffusion equation because a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in three-dimensional spherical coordinates and develop a three-level finite difference scheme for solving the heat transport equation in a microsphere. Stability of the scheme is proved in this dissertation. It is shown that the scheme is unconditionally stable. The scheme is then employed to investigate the temperature rise in a gold sphere subjected to a short-pulse laser. Numerical results are obtained for the cases that the laser irradiation is symmetric on the surface of the sphere, and the laser irradiation is from the top to a portion of the surface of the sphere

NUMERICAL INVESTIGATION OF THERMAL TRANSPORT MECHANISMS DURING ULTRA-FAST LASER HEATING OF NANO-FILMS USING 3-D DUAL PHASE LAG (DPL) MODEL

Ultra-fast laser heating of nano-films is investigated using 3-D Dual Phase Lag heat transport equation with laser heating at different locations on the metal film. The energy absorption rate, which is used to model femtosecond laser heating, is modified to accommodate for three-dimensional laser heating. A numerical solution based on an explicit finite-difference method is employed to solve the DPL equation. The stability criterion for selecting a time step size is obtained using von Neumann eigenmode analysis, and grid function convergence tests are performed. DPL results are compared with classical diffusion and hyperbolic heat conduction models and significant differences among these three approaches are demonstrated. We also develop an implicit finite-difference scheme of Crank-Nicolson type for solving 1-D and 3-D DPL equations. The proposed numerical technique solves one equation unlike other techniques available in the literature, which split the DPL equation into a system of two equations and then apply discretization. Stability analysis is performed using a von Neumann stability analysis. In 3-D, the discretized equation is solved using delta-form Douglas and Gunn time splitting. The performance of the proposed numerical technique is compared with the numerical techniques available in the literature

A FE-FD hybrid scheme for solving parabolic two-step micro heat transport equations in irregularly shaped three dimensional double -layered thin films exposed to ultrashort -pulse lasers

Multi-layer thin films are important components in many micro-electronic devices. These films are often used when a single film layer is insufficient to meet devices specifications. The continued reduction in component size has the side effect of increasing the thermal stress on these films and consequently the devices they comprise. Understanding the transfer of heat-energy at the micro-scale is important for thermal processing using a pulse-laser. Often, micro-voids may be found in processed devices. This is due to thermal expansion. Such defects may cause an amplification of neighboring defects resulting in severe damage and consequently the failure of the device. Thus a complete understanding of thermal dissipation and defects is necessary to avoid damage and to increase the efficiency of thermal processing. A hybrid finite element - finite difference (FE-FD) method has been developed for solving three dimensional parabolic two-step heat transport in irregular double-layered thin film exposed to ultrashort pulsed lasers. This scheme first discretizes the thin film system along the xy-plane by a finite element method. Then the z-direction is discretized via a weighted finite difference scheme. The two are combined into a numerical scheme which is then coded into a computer simulation. It is shown that the scheme is unconditionally stable with respect to the initial condition and the heat source. Three distinct numerical examples are studied. The first being a 0.05 μm gold thin film disk, with 1 mm diameter, atop a same-dimensioned chromium padding layer. This disk is exposed to an ultra-fast laser burst and the thermal properties are demonstrated. Secondly, the same thin-film disk array is exposed to a double burst laser pulse and the thermal properties examined. Finally the ultrashort laser is moved in a complete circle about the center of the double-layered thin disk and the thermal properties are examined. The outcome of this study provides an efficient and reliable numerical method for solving micro-scale heat transport equations, and gives a better understanding of the nature of heat transport in such a system. Also, the hybridization procedure offers a new way to examine three dimensional heat transport systems---one that utilizes the strengths of both the finite element and the finite difference methodologies. The research results have a significant impact on the development of short-pulse laser applications in structural monitoring of thin metal films, laser patterning of such films and laser synthesis and processing of thin film deposition

Exact and analytic-numerical solutions of lagging models of heat transfer in a semi-infinite medium

Different non-Fourier models of heat conduction have been considered in recent years, in a growing area of applications, to model microscale and ultrafast, transient, nonequilibrium responses in heat and mass transfer. In this work, using Fourier transforms, we obtain exact solutions for different lagging models of heat conduction in a semi-infinite domain, which allow the construction of analytic-numerical solutions with prescribed accuracy. Examples of numerical computations, comparing the properties of the models considered, are presented

A hybrid finite element-finite difference method for thermal analysis in a double-layered thin film

Thin film technology is of vital importance in microtechnology applications. For instance, thin films of metals, of dielectrics such as SiO2, or Si semiconductors are important components of microelectronic devices. The reduction of the device size to the microscale has the advantage of enhancing the switching speed of the device. The reduction, on the other hand, increases the rate of heat generation that leads to a high thermal load on the microdevice. Heat transfer at the microscale with an ultrafast pulsed-laser is also a very important process for thin films. Hence, studying the thermal behavior of thin films or of micro objects is essential for predicting the performance of a microelectronic device or for obtaining the microstructures. The objective of the research is to develop a numerical method for solving three-dimensional heat transport equations in a double-layered cylindrical thin film with microscale thickness. To this end, the three-dimensional heat transport equations are discretized using the finite element method for the x-y directions and the finite difference method for the z direction. Since the obtained scheme is implicit, a preconditioned Richardson iterative method is employed so that the systems of equations become only two block tri-diagonal linear systems with unknowns at the interface. Using a parallel Gaussian elimination procedure to solve these two block tri-diagonal linear systems, a domain decomposition algorithm for thermal analysis of a double-layered thin film is developed. The numerical procedure is employed to investigate the temperature rise and temperature distribution in a double-layered thin film with a cylindrical gold layer being on top of a cylindrical chromium padding layer. Numerical results are in good agreement with those obtained in previous research. The numerical method can be readily applied to the heat transport problem, where the shape of the film can be arbitrary in the x-y direction and where the film has multilayers

A finite difference method for studying thermal deformation in two-dimensional micro scale metal thin films exposed to ultrashort pulsed lasers

Ultrashort-pulsed lasers have been attracting worldwide interest in science and engineering because the lasers with pulse durations on the order of sub-picoseconds to femtoseconds possess capabilities in limiting the undesirable spread of the thermal process zone in a heated sample during material processing at the microscale. Prevention of thermal damage is an important factor for success of ultrashort-pulsed lasers in real applications. The thermal damage induced by ultrashort pulses is intrinsically different from that induced by long-pulse or continuous lasers. It occurs after the heating pulse is over and involves the shattering of thin metal layers (without a clear signature of thermal damage by excessive temperature) rather than the melt damage caused by high temperatures. In this dissertation, by replacing the displacement components in the dynamic equations of motion using the velocity components, and employing a staggered grid, we develop a finite difference method for studying thermal deformation in two-dimensional films exposed to ultrashort-pulsed lasers, where the thin films are a single-layered thin film and a double-layered thin film with perfectly interfacial thermal contact and imperfectly interfacial thermal contact, respectively. The method is obtained based on the parabolic two-step heat transport equations. It accounts for the coupling effect between lattice temperature and strain rate, as well as for the hot electron blast effect in momentum transfer. The developed methodology allows us to avoid non-physical oscillations in the solution. Such oscillations have been an intrinsic feature of most numerical method proposed so far in the context of problem of interest. The development of physical-based, numerical-oscillation-free methods for thermal analysis of thin metal films subjected to heating of ultrashort-pulsed lasers represents challenging tools at the forefront of this practically important area of research. This method is tested for its applicability by investigating the temperature rise and deformation in (1) a single-layered gold thin film, (2) a double-layered gold and chromium thin film with perfect thermal contact at the interface, and (3) a double-layered gold and chromium thin film with imperfect thermal contact at the interface. Results show that there are no non-physical oscillations in the solutions, and the method is promising

A numerical method to solve the two-step parabolic heat transport equations in a microsphere subjected to an ultrafast laser pulse

Heat transport at the microscale is the subject of intense investigation due to the growing need to fabricate microstructures for applications in nanotechnology. The need to control the spread of the thermal process zone has led to the development of high power short-pulse lasers. During thermal processing, impurities may form in the material. An amplification of the thermal energy around the impurities may result in severe damage occurring or in the failure of the thermal process. A thorough analysis of the way the impurities dissipates the thermal energy is therefore necessary to minimize the potential damage and optimize the thermal processing. The classical theory of heat diffusion, which is averaged over many grains, is inadequate in describing the transport phenomenon. Single energy equations developed to describe the transport phenomenon include a third-order mixed derivative with respect to space which makes them numerically inefficient. In this study, we will consider a microsphere subjected to an ultrafast laser pulse. The transport phenomenon is modeled by the two-step parabolic heat transport equations in three dimensional spherical coordinates. We will develop an energy estimate to establish the well-posedness of the problem, a three-level finite difference scheme to solve the transport equations, and prove that the finite difference scheme is unconditionally stable. The scheme will be applied to investigate the temperature rise in a gold sphere subjected to a short-pulse laser

NUMERICAL INVESTIGATION AND PARALLEL COMPUTING FOR THERMAL TRANSPORT MECHANISM DURING NANOMACHINING

Nano-scale machining, or Nanomachining is a hybrid process in which the total thermal energy necessary to remove atoms from a work-piece surface is applied from external sources. In the current study, the total thermal energy necessary to remove atoms from a work-piece surface is applied from two sources: (1) localized energy from a laser beam focused to a micron-scale spot to preheat the work-piece, and (2) a high-precision electron-beam emitted from the tips of carbon nano-tubes to remove material via evaporation/sublimation. Macro-to-nano scale heat transfer models are discussed for understanding their capability to capture and its application to predict the transient heat transfer mechanism required for nano-machining. In this case, thermal transport mechanism during nano-scale machining involves both phonons (lattice vibrations) and electrons; it is modeled using a parabolic two-step (PTS) model, which accounts for the time lag between these energy carriers. A numerical algorithm is developed for the solution of the PTS model based on explicit and implicit finite-difference methods. Since numerical solution for simulation of nanomachining involves high computational cost in terms of wall clock time consumed, performance comparison over a wide range of numerical techniques has been done to devise an efficient numerical solution procedure. Gauss-Seidel (GS), successive over relaxation (SOR), conjugate gradient (CG), d -form Douglas-Gunn time splitting, and other methods have been used to compare the computational cost involved in these methods. Use of the Douglas-Gunn time splitting in the solution of 3D time-dependent heat transport equations appears to be optimal especially as problem size (number of spatial grid points and/or required number of time steps) becomes large. Parallel computing is implemented to further reduce the wall clock time required for the complete simulation of nanomachining process. Domain decomposition with inter-processor communication using Message Passing Interface (MPI) libraries is adapted for parallel computing. Performance tuning has been implemented for efficient parallelization by overlapping communication with computation. Numerical solution for laser source and electron-beam source with different Gaussian distribution are presented. Performance of the parallel code is tested on four distinct computer cluster architecture. Results obtained for laser source agree well with available experimental data in the literature. The results for electron-beam source are self-consistent; nevertheless, they need to be validated experimentally

A finite difference method for studying thermal deformation in a three-dimensional microsphere exposed to ultrashort-pulsed lasers

Ultrashort-pulsed lasers with pulse durations on the order of sub-picoseconds to femtoseconds possess the capabilities in limiting the undesirable spread of the thermal process zone in a heated sample which have been attracting worldwide interest in science and engineering. Success of ultrashort-pulsed lasers in real application relies on: (1) well characterized pulse width, intensity and experimental techniques; (2) reliable microscale heat transfer models; and (3) prevention of thermal damage. Laser damage by ultrashort-pulsed lasers occurs after the heating pulse is over since the pulse duration time is extremely short and the heat flux is essentially limited to the region within the electron thermal diffusion length. In contrast with long-pulse laser, laser damage is caused by melting temperature resulting from continuous pulse of energy. This dissertation investigates the mathematical model of heat transport phenomenon in a 3D micro-sphere exposed to ultrashort-pulsed lasers and presents a numerical method for studying thermal deformations. The method is obtained based on the parabolic two-step model and implicit finite difference schemes on a staggered grid. It accounts for the coupling effect between lattice temperature and strain rate, as well as for the hot electron blast effect in momentum transfer. In particular, a fourth-order compact scheme is developed for evaluating those stress derivatives in the dynamic equations of motion. It should be pointed out that micro-spheres are considered because they are of interest related to micro resonators in optical applications, such as ultra-low-threshold lasing, sensing, optoelectronic microdevices, cavity quantum electrodynamics and their potential in quantum information processing. The numerical method is tested for its applicability by investigating the temperature rise and deformation in five examples, which are (1) a portion of the upper hemisphere is irradiated by a single-pulse laser, (2) portions of both the upper hemisphere and the lower hemisphere are irradiated by a single-pulse laser, (3) the upper hemisphere is irradiated by a single-pulse laser, (4) a portion of the upper hemisphere is irradiated by a double-pulse laser, and (5) portions of both the upper hemisphere and the lower hemisphere are irradiated by a double-pulse laser. Results show that no non-physical oscillations appear in the solutions and the micro-sphere expands when it is irradiated by ultrashort-pulsed lasers

The Zoo of Non-Fourier Heat Conduction Models

The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or time lags become dominant and the memory or/and spatial non-local effects significant -- in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid state laser medium at the high pump density or under the ultra-short pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from $10^{-8}$ to $10^{-14}$ sec, however, it could be as high as 1 sec in the degenerate cores of aged stars and its reported values in granular and biological objects varies up to 30 sec. The paper considers numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory (hereditary materials, including fractional hereditary materials) or/and spatial non-locality, i.e. materials with non-homogeneous inner structure