The Pennsylvania State University. The Graduate School. College of Engineering FLUCTUATIONS IN THERMAL FIELD OF WORKPIECE DUE TO MULTIPLE PULSES

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1 The Pennsylvania State University The Graduate School College of Engineering FLUCTUATIONS IN THERMAL FIELD OF WORKPIECE DUE TO MULTIPLE PULSES IN ULTRA-HIGH PRECISION LASER NANO-MACHINING A Thesis in Engineering Science and Mechanics by Hana H. Krechel 2014 Hana H. Krechel Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December 2014

2 ii The thesis of Hana H. Krechel was reviewed and approved* by the following: Edward W. Reutzel Associate Professor of Engineering Science and Mechanics Thesis Advisor Vladimir V. Semak Associate Professor of Engineering Science and Mechanics Timothy F. Miller Energy Science and Power Systems Division Head, Applied Research Laboratory Graduate Faculty in Mechanical Engineering Judith A. Todd Professor of Engineering Science and Mechanics P. B. Breneman Department Head *Signatures on file in the Graduate School

3 iii ABSTRACT Laser drilling and nano-machining have become widely used techniques for industrial manufacturing. Ultra-short pulse lasers, in the picosecond to femtosecond range, are desirable for micron and sub-micron level precision. However, based on experimental observations and recent research, the highly precise and consistent drill depth required in some applications has not been achieved at this level. Despite significant efforts to keep all material and machining properties stable, micrometer level pulse-topulse fluctuations are apparent. The work presented here attributes these fluctuations to a fluctuating temperature field. It is hypothesized and confirmed in simulation that fluctuations in drill depth in surface texturing and surface machining are due to transient temperature effects from multiple pulses despite maintaining all machining properties constant. The effects are numerically simulated and studied by extending the current single-pulse model. It is observed that transient temperature effects are a significant contributing factor to observed fluctuations since the material and machining properties have been modeled as perfectly stable. The temperature field itself creates instabilities for multiple laser pulses, and can be correlated to instabilities in drill depth and melt depth. This unexplored area of research can be of benefit to industry. Once fluctuations have been fully modeled and understood in relation to drill depth and melt depth, it will be possible to refine the precision and consistency of high repetition rate ultra-short pulse laser machining at the nanometer level.

4 iv TABLE OF CONTENTS List of Figures.... v Nomenclature....vi Introduction....1 CHAPTER 1. Transient Heat Transfer Processes..3 Transient Heat Conduction... 3 Convective Cooling.. 4 CHAPTER 2. Thermal Processes in Laser-Material Interaction Energy Absorption....5 Conductive Heat Transfer Evaporative Cooling. 6 Critical Temperature. 7 CHAPTER 3. Description of Modeling Process....9 Finite Difference Approximation Simulation of Laser Energy Deposition.. 10 Assumptions Sample of Results Obtained CHAPTER 4. Thermal Instabilities in Laser Nano-Machining Description of Instabilities.. 17 Results Found.. 23 One Pulse per Each Location.23 Two Pulses per Each Location...24 Four Pulses per Each Location..25 Factors Affecting Fluctuations Changing Substrate Thickness...26 Changing Pulse Repetition Rate Changing Heat Diffusion Coefficient CHAPTER 5. Conclusion Summary and Discussion Works Cited. 38 Appendix A: Nontechnical Abstract Appendix B: Time Step Independence....40

5 v LIST OF FIGURES Figure 3-1: Simulation Model Figure 3-2: Temperature versus time for twelve pulses of a laser Figure 3-3: Maximum temperature versus time for twelve pulses of a laser Figure 4-1: Temperature versus depth for a single pulse of a laser Figure 4-2: Temperature versus depth for the first and second pulse of a laser..19 Figure 4-3: Change in temperature versus depth. 20 Figure 4-4: Heat deposition for the first pulse of a laser. 21 Figure 4-5: Heat deposition for two pulses of a laser at the same location.21 Figure 4-6: Heat deposition for three pulses of a laser at the same location...21 Figure 4-7: Heat deposition for four pulses of a laser at the same location Figure 4-8: Heat deposition for five pulses of a laser at the same location.22 Figure 4-9: Maximum temperature reached at the point of laser pulse, one pulse per location Figure 4-10: Maximum temperature reached at the point of laser pulse, two pulses per location.. 24 Figure 4-11: Maximum temperature reached at the point of laser pulse, four pulses per location..25 Figure 4-12: Thickness of.01 meters Figure 4-13: Thickness of.02 meters Figure 4-14: Thickness of.04 meters Figure 4-15: Pulse Repetition Rate of ~30 khz Figure 4-16: Pulse Repetition Rate of ~40 khz Figure 4-17: Pulse Repetition Rate of ~50 khz Figure 4-18: Heat Diffusion Coefficient of.05 centimeters squared per second Figure 4-19: Heat Diffusion Coefficient of.1 centimeters squared per second.. 31 Figure 4-20: Heat Diffusion Coefficient of.2 centimeters squared per second Figure 5-1: Surface temperature of 4 laser pulses in the same location Figure 5-2: Background temperature of 4 laser pulses in the same location Figure 5-3: Maximum temperature of 4 laser pulses in the same location..34 Figure B-1: Time Steps of.2 µs...40 Figure B-2: Time Steps of.02 µs

6 vi NOMENCLATURE A C p h H I i J K k n n Q ab Q in Q r t T background T critical T env T s T u x y z Surface Area Specific Heat for Constant Pressure Heat Transfer Coefficient Enthalpy Laser Beam Intensity Location Step Mass Flux Thermal Conductivity Imaginary Part of the Refractive Index Real Part of the Refractive Index Time Step (in Finite Difference Equations) Thermal Energy Absorbed Thermal Energy input Thermal Energy Stability Condition Time Background Temperature Critical Temperature Temperature of the Environment Surface Temperature Temperature Arbitrary Function Width Length Depth

7 vii z cr α γ ΔH v ΔT Δx Δy Δz ρ Critical Depth Arbitrary Constant Absorption Coefficient Change in Enthalpy for Liquid to Vapor Transition Change in Temperature Change in x Change in y Change in z Density

8 1 INTRODUCTION Ultra-short pulse laser drilling and nano-machining have become widely used techniques for industrial manufacturing. Among the many advantages of laser drilling over mechanical or chemical drilling are high speed processing and micron (and sub-micron level) precision. However, researchers have reported that the desired level of precision is not always achieved when multiple pulses are required. Refinements in the efficiency and precision of the laser drilling and laser micromachining process are of great interest to both industry and academia. The work presented here proposes a solution to further improve upon the pulse-to-pulse precision of laser drilling and micromachining in industry. Micrometer level pulse-to-pulse fluctuations in drill depth and melt depth have been experimentally observed using picosecond to femtosecond lasers. In general, high quality laser machining becomes a discussion of hole quality, which scientist and engineers go to great lengths to optimize, using stabilization measures in their drilling strategies [1]. Improvements in efficiency and productivity of laser-material interaction has been attributed to proper control of the laser beam and the workpiece [1]. In cases for both long and short pulses, any small inaccuracies in pulse-to-pulse drill depth and melt production have been attributed to instabilities in machining properties or imperfections in the material itself, along with instabilities in other physical properties such as melt flow or vaporization front [2, 3]. However, the research presented here demonstrates that there is another contributing factor to inaccuracies for ultra-short (picosecond to femtosecond range) pulses: a fluctuating temperature field. Existing models of laser-material interactions, such as the Two-Temperature Model (TTM), have been problematic when it comes to predicting melt at higher laser intensities [4, 5]. Significantly more material is removed than predicted. Past research in this area has proposed a correction to the TTM that accounts for this excessive removal of material. Essentially, the ponderomotive force on electrons in the skin depth region accelerates the electrons deeper into the metal, where energy, in the form of heat, is deposited [4, 5]. Inclusion of the ponderomotive force correction to the TTM accurately predicts drill depth and melt depth of laser nano-machining and has been fully comprehended and modeled for a single laser pulse [4, 5]. The research presented here further enhances this improved model by expanding its application to account for multiple laser pulses. This newly-created model simulates the temperature field that is generated when material is irradiated by multiple pulses (as opposed to a single pulse) on a longer time scale (as opposed to the time scale for a single pulse) and in three dimensions. There are multiple heating and cooling effects that define the temperature near the source point of a laser pulse. The input of energy from the laser, conductive heat transfer, convective heat transfer, and evaporative cooling are some of the competing heat transfer effects that influence the temperature field. This simulation shows that the

9 2 thermal field itself near the laser interaction zone at the surface of the workpiece is not steady when multiple pulses are applied to the same point. Fluctuations in the thermal field are shown to be dependent on factors such as thickness of the substrate, heat diffusion coefficient of the workpiece, and pulse repetition rate of the laser. The results presented here will help to improve the precision provided by ultrashort pulse laser nano-machining in applications including micro-machining during production of microelectronic devices like integrated circuits and in micro-drilling of cooling holes in turbine blades. Simulation of thermal field variation that occurs when machining with multiple laser pulses may be used to help optimize and stabilize industrial laser processing. Although equipment stability and spatial variation in material properties will play a role in material removal during laser processing, understanding fluctuations of the thermal field and fully calculating the effect that it has on melt depth and material removal will lead to improved precision in the field of laser drilling and nano-machining.

10 3 Chapter 1 TRANSIENT HEAT TRANSFER PROCESSES In their most basic form, transient heat transfer equations can be used to describe heating and cooling as a function of time and space. The fundamental thermodynamic processes outlined in this chapter are necessary for understanding and interpreting the more specific laser heating and cooling processes described in Chapter 2. Transient Heat Conduction Internal heat conduction is defined as the transfer of energy within a body due to a temperature gradient, and heat spontaneously flows from a hotter body to a cooler body. A body desires to reach thermal equilibrium over an extended period of time. In non-steady-state conduction, or transient conduction, temperature can be determined by solving equations that are functions of time and space coordinates [6]. Heat conduction can be described using the heat equation, or Fourier s law. Equation 1.1 represents an arbitrary function, u, represented by Fourier s law: In this equation, α is a constant, and x, y, z, and t are space and time coordinates. When this equation is applied to heat transfer, it is generally written as: (1.1) In Equation 1.2, T is time and space dependent temperature, α is thermal diffusivity, and 2 denotes the Laplace operator. This differential equation can be readily solved numerically through computer simulations. (1.2)

11 4 Convective Cooling Convection is a general term for heat transfer through a moving fluid. It is the mode of energy transfer between a solid surface and the adjacent liquid or gas [6]. Convective cooling can be described using Newton's law of cooling when h, the heat transfer coefficient, is assumed constant, as it is in this simulation. Newton s law states that the rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings. In Equation 1.3, Q is thermal energy lost through convection, A is the heat transfer surface area, and h is the convective heat transfer coefficient, assumed independent of temperature. Convective cooling is a necessary boundary condition and can also be solved numerically through computer simulations. These theories and equations are well-known and are utilized in the formulation of boundary conditions, explained in Chapter 3. (1.3)

12 5 Chapter 2 THERMAL PROCESSES IN LASER-MATERIAL INTERACTION Modern theories of laser-material interaction can describe thermodynamic interactions in enough detail to explain the interaction of the laser energy with the electrons and ions [4, 5]. Relaxation times for electrons and ions are in the picosecond range, and would require much smaller time steps in order to accurately simulate the physical phenomena [7]. However, local thermodynamic equilibrium can be established within the workpiece (over space and time) when the diffusion time scale of the system is much larger than the relaxation time scale of relevant energy carriers. In this study, the time steps of the numerical simulation are in the nanosecond (or longer) range, so that thermal equilibrium is achieved. Therefore, it is appropriate to use Fourier heat conduction to model the temperature interactions of the laser. Energy Absorption The energy deposited in the workpiece due to the pulse of a laser is highly dependent on the material s optical properties. Consider Q ab, the volumetric thermal energy intensity absorbed by the material [7] : In Equation 2.1, I is the laser beam intensity, γ (gamma) is the absorption coefficient of the workpiece, and x, y, z, and t represent the space and time coordinates. The surface reflectivity, R, is given by [7] : (2.1) In Equation 2.2, n is the real part of the refractive index and k is the imaginary part. In this equation, n and k are known to be dependent on temperature. (2.2)

13 6 In a more general equation, the volumetric energy absorption is given with a temperaturedependent absorption coefficient, where T s is the surface temperature [7] : These equations assume uniform material properties, with the laser-material interaction occurring in a vacuum. Scattering occurs in actual applications, but is assumed to be negligible here in order to simulate ideal laser machining conditions. (2.3) Conductive Heat Transfer As noted previously, for time steps longer than characteristic relaxation times of energy carriers (as modeled in this simulation), Fourier heat conduction sufficiently describes the heat transfer processes. Fourier s law was given in general terms in the previous chapter and is given here specifically for the heat conduction equation. Hence, the transient (time-dependent) temperature field can be solved through the heat conduction equation [7] : In Equation 2.4 ρ, C p, T, and K represent density, specific heat for constant pressure, temperature, and thermal conductivity of the substrate. Although technically these properties are temperature dependent, in this work, they are assumed constant to develop approximate solutions. This is assumption generally accepted in thermodynamics calculations [6]. The energy deposited in the workpiece due to the pulse of a laser is represented by Q ab. (2.4) Evaporative Cooling In laser-material thermal interactions, evaporative cooling is another significant heat transfer phenomenon that must be accounted for. As a material is heated above the boiling point, the material is transformed into a vapor. As evaporation occurs, the material that is removed from the interaction zone is at a higher temperature than the material that remains and effectively removes a portion of the initially deposited energy. Additionally, vaporization absorbs process energy, and thus the material left behind is

14 7 at a lower temperature. In laser-material interaction this accounts for a significant amount of surface cooling and surface temperature saturation. The boundary condition that describes evaporative cooling is given in Equation 2.5 [8]. This boundary condition applies to the top surface, and has a significant effect as the laser deposits high amounts of energy at the surface. Here, K is thermal conductivity, T/ z z = 0 is the temperature gradient at the surface, J is mass flux due to evaporation, and ΔH v is change in enthalpy for liquid to vapor transition. Essentially, the larger the temperature gradient due to laser energy deposited into material, the greater the energy loss due to material evaporation. Therefore the higher the initial temperature reached during the laser pulse, the greater the cooling effect due to evaporative cooling. (2.5) Critical Temperature The critical point and the so-called critical temperature is a well-defined concept in thermodynamics, and is defined as the point at which saturated liquid and saturated vapor states are equal [6]. However, a direct application of this concept to laser-material interactions is not straightforward. The difficulty arises due to evaporation that occurs as a result of ultra-short pulse ultra-high intensity lasermaterial interaction. Evaporation resulting from a pulsed laser is known to expel both large and small clusters of material [9]. This process of laser ablation has been extensively investigated both experimentally and theoretically, and ongoing research is attempting to develop an understanding of the transition from a superheated liquid to exploded material [10]. When the surface of a material reaches a temperature higher than the critical temperature, at sufficiently high laser intensity, ablative material removal is induced. This process is commonly known as phase explosion or explosive boiling. Explosive boiling has been described as the transition from a superheated liquid to a mixture of liquid and vapor, in which the material heated above the critical temperature is expelled from the surface in a mixture of vapor and liquid droplets. The plume created due to this process is a mixture of individual atoms/molecules, clusters of material, and liquid droplets.

15 8 Once the critical temperature is reached, the near-surface of the material becomes overheated, and is no longer thermodynamically stable. This leads to phase explosion of the material, and the material is expelled from the surface. Therefore, for the purpose of this work, the critical temperature of the material is viewed as the point at which the material can be removed from the simulation. Specifically, for longer time scale, multi-pulse laser-material interaction modeling, it can be assumed that material heated above the critical temperature undergoes a phase explosion and is expelled from the surface, and may thus be discounted from further calculations in the model. Thus, the laser-material interaction process can be simulated on a macro scale without detailed representation of molecular and fluid dynamics.

16 9 Chapter 3 DESCRIPTION OF MODELING PROCESS Finite Difference Approximation Finite difference methods provide an approximate solution for a continuous partial differential equation, in this case, the heat equation. This technique involves creation of a mesh of points, called nodes; where at each node a solution is computed [11]. The nodes are separated in space by a designated Δx, Δy, and Δz, creating computational blocks, and are separated in time by a designated Δt. To compute a solution, it is required to specify initial conditions at t = 0 and boundary conditions for all faces, edges, and corners. In this case, convective cooling boundary conditions are applied, and heat conduction within the material was calculated. The evaporative cooling boundary condition was also applied to the top surface. There are various numerical methods that can be used to find approximate solutions to the heat equation [11]. The Forward-Time Centered-Space method is illustrated using the one-dimensional heat equation as an example, but can be easily expanded to three dimensions. Consider the one-dimensional heat equation where 0 x L, and t 0. This equation describes transient heat conduction in a slab of material with a thickness of L. The Forward-Time Centered-Space approximation to the heat equation is given in Equation 3.2, where i represents the location step and n represents the time step [11]. (3.1) A so-called stability condition in numerical analyses can be used to ensure that computational errors do not grow within the sequence of the numerical procedure [11]. For this method, the stability condition is [11] : (3.2) (3.3)

17 10 Note that in three dimensions, this stability condition also applies to y and z. Application of this condition ensures that as Δt grows, errors from any source (rounding, truncation, etc.) do not grow [11]. Numerical errors generated when applying this numerical solution are proportional to the time step and the square of the space step. It should also be noted that the Forward-Time Centered-Space method is an explicit solution method. An explicit solution method calculates the state of a system at a later time based on the current (known) state of a solution. In comparison, implicit solution methods calculate the state of a system based on the current state and a later state. Implicit solution methods require extra computation in each iteration, but can utilize larger time steps. When using an explicit solution method, accuracy and numerical stability are maintained by using small time steps; however, computation time is often reduced because explicit methods require one fewer computation for each time step [11]. A solution for the thermal field can be developed using the Forward-Time Centered-Space explicit solution method to the Fourier equation and applying appropriate boundary conditions. The thermal energy from a laser pulse was applied as an energy flux on the top surface at the designated time and location. Evaporative cooling (temperature dependent) was included on the top surface boundary condition. Simulation of Laser Energy Deposition In order to model the input of energy from the laser, Equation 2.1 is rewritten as: In Equation 3.4, Δx, Δy, and Δz describe the size of one computational block. Volumetric thermal energy absorbed is related to change in temperature in the computational cell through the Equation 3.5: (3.4) In Equation 3.5, C p is specific heat at constant pressure and ρ is density. As briefly described above and further explained in the next chapter, during the laser process, any material heated above the critical temperature is expelled from the surface, leaving material with a surface temperature equal to the critical temperature, and exponentially cooler temperature with increasing depth. This specific depth at which the temperature of the material is the critical temperature (prior to material expulsion) will be referred to as z critical, or z cr. Material that is removed from the surface (in the range 0 < z (3.5)

18 11 < z cr) does not contribute to the heating process in the remaining material, since energy contained in the expelled material is removed from the surface, and is thus not absorbed into the material. Therefore, in order to simulate the input of thermal energy from the absorbed laser energy, Q in, it must be averaged over the height of one computational block, starting at the critical depth. Hence, This average over the computational block, beginning at the critical depth, was used to calculate change in average temperature of the computational block due to volumetric thermal energy absorbed from the pulse of a laser. In other words, the very high thermal gradients that would actually be present near the surface of the material are averaged over the size of the computational block for the purpose of this simulation. This smearing effect can be reduced when simulation is performed with smaller Δx, Δy, and Δz steps; however, maintaining large computational blocks provides a means to reduce the computational cost when modeling large scale thermal effects. Figure 3-1 describes the simulation model. The material is modeled with a specified size, expressed by x, y, and z. The values for x and y were in the range of 1 to 2 meters, and 0.01 to 0.05 meters for depth, z. This piece of material was then broken down into a mesh of computational blocks of a specified size Δx, Δy, and Δz. These variables ranged from to meters. The input of thermal energy from the laser was modeled as instantaneous in one computational block at the specified time and location. (3.6)

19 12 Figure 3-1: Simulation Model Assumptions There were several assumptions made in creating this model. These assumptions are not believed to degrade the qualitative correctness of the model, but are recognized to result in quantitative inaccuracies. All simplifications and assumptions are prescribed to maintain qualitative integrity of the process over the long time steps used in this model. First, consider the simulation of a laser pulse and the laser energy deposition, described in the preceding section. In order to model the thermal energy added from a laser, the beam was modeled as being square (the size of one computational block). The heat input was also assumed to be delivered instantaneously. These assumptions do not affect the qualitative accuracy of this model because characteristic relaxation time for energy transfer of laser pulses is in the picosecond range [7]. Heat transfer occurring at this time scale need not be considered when time steps are in the nanosecond and longer range.

20 13 Sample of Results Obtained The simulation calculated the temperature of every computational block in the workpiece at every time step. In order to interpret these results, research focused almost exclusively on the temperature near the point at which the laser interacted with the material. Therefore, for each time step, the simulation produced the temperature where the laser was located. When the laser translated to a different location on the surface, the temperature was calculated at that new location. Therefore, results are given in time (as the time step moved) and with a moving reference frame (following the laser). For example, consider a laser translating across the surface of the material. This laser pulsed one time in each computational block in a straight line in the y-direction. The resulting temperatures are shown in Figure 3-2. This graph shows temperature in the vertical axis and time in the horizontal axis. These temperatures represent the temperature averaged over the surface computational block after the energy from the expelled material has been eliminated from the simulation. Since the laser is moving between pulses, a second axis is included showing the Δy location step. In this particular simulation, the laser traversed in a straight line along the y direction. Each pulse occurred at Δy distance away from the previous pulse; therefore, the temperature in Figure 3-2 is plotted at that new location. As a general rule, the temperature plotted in all subsequent graphs is the temperature at the surface of the material wherever the laser beam is currently located.

21 Surface Temperature (K) Δy Location Step (.5mm) 14 Surface Temperature of 12 Laser Pulses E E E E E E E-03 Time (s) 0 Figure 3-2: Temperature versus time for twelve pulses of a laser Figure 3-2 shows twelve pulses of a laser, and each pulse of the laser occurred in a straight line across the surface of the workpiece, one pulse occurring every Δy =.5 millimeters. The temperature (i.e. temperature averaged over the computational block) peaks during the laser pulse, and then the temperature cools down. The peak occurs at the new location of the laser pulse, and then that new location cools down as well before the laser moves to the next location. In order to achieve the overall objective of examining the stability of the thermal field, these temperature graphs were examined by focusing attention on the maximum temperature reached for each pulse. Maximum temperature is important, especially when making the results useful to industry. For example, the maximum temperature can be used to calculate drill depth or amount of material removed [5].

22 Maximum Temperature (K) Δy Location Step (.5mm) 15 In order to more closely examine the maximum temperature achieved, Figure 3-3 plots the maximum temperature reached for the same 12 laser pulses. Again, these maximum temperatures are moving in both time and space. Time is plotted on the horizontal axis, and the location of each maximum is determined by the location of the laser pulse, plotted in the Δy location step function. Note the peak temperature rises steadily throughout this simulation. Maximum Temperature of 12 Laser Pulses E E E E E E E-03 Time (s) Figure 3-3: Maximum temperature versus time for twelve pulses of a laser

23 16 In the next chapter, the model was applied to a case in which a laser is pulsed multiple times in the same location prior to moving to the next location. In order to examine the stability of the thermal field, research focused on the maximum temperature reached in the laser-material interaction zone. Therefore, graphs for maximum temperatures reached are plotted in the same manner, in the sense that the maximum surface temperature was calculated at the beam (averaged over the computational block after the energy from the expelled material has been removed), and when the beam moves (after a given number of pulses in the same location), the temperature was calculated at that new location. For clarity, the graphs shown in Chapter 4 do not include the second axis for Δy location; however, it is understood, as a general rule, that maximum temperature plotted is at the location of the laser beam.

24 17 Chapter 4 THERMAL INSTABILITIES IN LASER NANO-MACHINING Description of Instabilities A model for multi-pulse laser drilling or micromachining has been developed, and simulations of multiple pulses along the surface of a material have revealed thermal fluctuations. These thermal fluctuations become more interesting when multiple pulses of a laser are repeated at one location on the surface of the material. The temperature at this spot is not uniform at each pulse, nor is it consistently increasing or decreasing. In fact, thermal instabilities create fluctuations in the temperature. These fluctuations are the results of competing heating and cooling effects. Before any laser irradiation, the temperature of the workpiece is at the prescribed background temperature. The first pulse of the laser deposits energy in the material. The surface temperature of the material rises to a temperature higher than the original background temperature, and temperature exponentially decreases as the depth, z, increases. This is shown in Figure 4-1, a graph of temperature versus depth. As described in Chapter 2, when a material is heated above the critical temperature, a thermal explosion occurs, and the material is expelled. The remaining material has a lower temperature, starting at the critical temperature and exponentially decreasing with depth. The first laser pulse generates a rise in temperature, relative to background temperature. This pulse deposits a certain amount of energy throughout the volume of the computational area, and the change in temperature is proportional to this volumetric energy deposition.

25 18 Figure 4-1: Temperature versus depth for a single pulse of a laser The shaded region in Figure 4-1 represents the energy deposited in the material. Since material in the range 0 < z < z cr is expelled due to thermal explosion above critical temperature, that energy is not deposited in the material. After cooling, the deposited energy leads to an increase in substrate temperature (as compared to the background temperature prior to the first pulse). Now, consider a second pulse at the same location, which irradiates the substrate that is at a temperature greater than the original background temperature. As seen in Figure 4-2, the graph of temperature versus depth for the second pulse is the same as the first, except slightly shifted up due to the increase in background temperature. Since critical temperature remains the same, total energy deposited in the material is reduced, because more energy is lost due to material expulsion resulting from the thermal explosion above the critical temperature. Less energy deposited in the material for the second pulse leads to a decrease in maximum surface temperature reached (as compared to the first pulse). Background temperature influences the amount of energy deposition for a laser pulse.

26 19 In Figure 4-2, the shaded region represents the energy deposition of the second pulse, which, when compared to Figure 4-1, is shown to be less than the first pulse. Figure 4-2: Temperature versus depth for the first and second pulse of a laser This effect can also be explained by considering the change in temperature, ΔT, which is a reflection of absorbed energy from the laser pulse. Equation 4.1 represents the temperature at the surface of the material. The temperature, T, can be found by summing the change in temperature and the background temperature. At the point of the surface immediately after the pulse, this is critical temperature, since all material at a temperature higher than critical temperature has been expelled. Hence, at the new surface: (4.1) (4.2)

27 20 Since background temperature increases for the second pulse (T bkg2 > T bkg1), this leads to a smaller change in temperature as compared to the first pulse. Therefore, the second pulse of the laser deposits less energy into the material than the first pulse at the same location, because the initial temperature of the substrate prior to the laser pulse has increased, as shown in Figure 4-3. Figure 4-3: Change in temperature versus depth Background temperature influences the amount of laser energy imparted to the substrate. A higher background temperature results in lower laser energy deposition. This effect would continue for subsequent pulses, except that the heat conduction and surface convection serve to transport heat from the interaction zone and lower the substrate temperature near the surface. As described in prior chapters, the heat from the laser diffuses radially through the material, away from the source point of the laser beam. See Figures 4-4 through 4-8.

28 21 Figure 4-4: Heat deposition for the first pulse of a laser If the second pulse is close behind the first pulse, the heating effects from the first pulse remain, causing further penetration of the thermal field, diffusing radially outward from the source point of the laser, as can be seen in Figure 4-5. Figure 4-5: Heat deposition for two pulses of a laser at the same location Eventually, this heat reaches the bottom surface of the material and starts to conduct and dissipate further radially outward from the source point of the laser. The top and bottom boundary provide a convective cooling boundary condition. Later repetition of pulses in the same location allow more time for the heat to diffuse radially away from the source point, shown in Figures 4-6 and 4-7. Figure 4-6: Heat deposition for three pulses of a laser at the same location

29 22 Figure 4-7: Heat deposition for four pulses of a laser at the same location Soon after, heat dissipates far enough away from the source point of the laser that it leads to a change in background temperature at the surface of the workpiece. This effect is observed in later pulses repeated at the same location. See Figure 4-8. Figure 4-8: Heat deposition for five pulses of a laser at the same location Ultimately, heat conduction and diffusion lead to a change in background temperature, which leads to a change in the amount of laser energy imparted to the substrate. However, the heat conduction and dissipation effect described above occurs with a significant time delay (relative to the laser pulse), due to the time that is required for heat to move through the material. Heat conduction and dissipation throughout the material have an effect in later pulses as more pulses are repeated and more time passes. Since the two processes described in this section are occurring at different rates, temperature fluctuations are seen, highly dependent on the material properties and material geometry.

30 Maximum Temperature (K) 23 Results Found Figures 4-9 through 4-11 represent the maximum temperature (averaged over the computational block at the surface, with volume of 4x10-4 cm 3 ) that is reached at the source point of the laser pulse. The pulse duration and pulse energy ranges used in the simulations were typical of industrial ultra-short pulse laser machining. The temperatures are graphed in both time and space as described above, though the step function showing y-location is emitted for clarity. The laser deposits energy in one computational block then moves Δy distance along the y-axis to the next computational block. Four cases are presented below, with increasing number of pulses occurring at each location before the laser moves to the next location. One Pulse per Each Location (Figure 4-9) This baseline simulation example shows that if no pulses are repeated at the same location, no thermal fluctuations at the surface of the material were discovered. The temperature at the location of the laser beam increases slightly for each pulse due to general heating throughout the material from the pulses before it Maximum Surface Temperature for Laser Pulse Repeated 1 Time at Each Location E E E E E E E E-03 Time (s) Figure 4-9: Maximum temperature reached at the point of laser pulse, one pulse per location

31 Maximum Temperature (K) 24 Two Pulses per Each Location (Figure 4-10) As described above, the second pulse repeated at the same location reaches a lower maximum temperature at the surface than the first pulse. This is because the higher background temperature for the second pulse leads to less energy deposition as compared to the first pulse. Note that this graph is also plotted in both time and space for each pair of pulses in the same location; Figure 4-10 follows the laser as it pulses twice in one location, then moves forward one space step along the y-axis. The step function showing y location is omitted, but it is understood that the maximum temperature plotted is at the location of the laser for the two pulses in the same location. The first pulse is the higher temperature, and the second pulse is the lower temperature. One example pair of pulses in the same location is circled for clarity. 466 Maximum Surface Temperature for Laser Pulse Repeated 2 Times at Each Location E E E E E E E E-03 Time (s) Figure 4-10: Maximum temperature reached at the point of laser pulse, two pulses per location

32 Maximum Temperature (K) 25 Four Pulses per Each Location (Figure 4-11) As described above, the more pulses that are repeated at the same location eventually lead to a change in background temperature due to heat conduction and diffusion, which leads to a change in energy deposition. In this example, heat conduction causes an effect by the fourth pulse. Again, note that this graph is plotted in both time and space for the four pulses in the same location; Figure 4-11 follows the laser as it pulses four times in one location, then moves forward to the next laser pulse location along the y-axis. The step function showing y location is omitted, but it is understood that the maximum temperature plotted is at the location of the laser for the four pulses in the same location, then moves Δy along the y-axis before the next four pulses. The maximum temperature reached at the surface of the material at the location of the laser pulse is highest after the first pulse, then lower after the second pulse, slightly lower after the third pulse, then slightly higher after the fourth pulse. An example group of four pulses in one location is circled for clarity Maximum Surface Temperature for Laser Pulse Repeated 4 Times at Each Location E E E E E E E E-03 Time (s) Figure 4-11: Maximum temperature reached at the point of laser pulse, four pulses per location

33 Temperature (K) 26 Factors Affecting Fluctuations All results given in this section are plotted as the maximum surface temperature at the source point of the laser (averaged over that computational block and after excluding energy removed in the expelled material) for four laser pulses repeated in the same location. Changing Substrate Thickness Substrate thickness is the first important factor affecting thermal fluctuations. While keeping all other laser processing properties the same, it was predicted that the thinner the workpiece, the faster the heat will conduct to the bottom boundary (a convective cooling boundary), and the sooner an effect due to conduction would be observed. Such an effect has been seen through simulation of multiple thicknesses of material. Decreasing the thickness of the workpiece decreases the time it takes to observe an effect in maximum temperature at the surface due to heat dissipation (due to its effect on background temperature). An effect from heat conduction and dissipation can be seen in the earlier repetitions of pulses. While maintaining all other machining properties, the thickness of this sample was varied from 0.01 to 0.02 to 0.04 meters. Four pulses were repeated in each location. Results found appear to support the hypothesis. An example group of four pulses in one location is circled for clarity Thickness of.01 meters E E E E E E E E-03 Time (s) Figure 4-12: Thickness of.01 meters

34 Temperature (K) Temperature (K) Thickness of.02 meters E E E E E E E E-03 Time (s) Figure 4-13: Thickness of.02 meters Thickness of.04 meters E E E E E E E E-03 Time (s) Figure 4-14: Thickness of.04 meters

35 Temperature (K) 28 As can be seen, substrate thickness has an effect on thermal fluctuations, specifically in the later repetitions of pulses. This data suggests that thinner material demonstrates an earlier effect (a fluctuation to a higher temperature) due to heat conducting to the bottom convective cooling boundary more quickly. In Figure 4-12, an effect can be seen by the third or fourth pulse. This effect becomes less prevalent and eventually non-existent when the thickness of the material is increased. Changing Pulse Repetition Rate The pulse repetition rate has an effect on thermal fluctuations. While keeping all other machining properties the same, the pulse repetition rate was varied. Higher pulse repetition rates exaggerate the cooling effect between the first and second pulses due to varying background temperatures. When there is less time for the surface of the material to cool between pulses, a higher background temperature leads to a decrease in energy deposited to the substrate. While maintaining all other laser processing properties, the pulse repetition rate was varied from approximately 30 khz to 40 khz to 50 khz. Four pulses were repeated in each location. Results appear to support the hypothesis. An example group of four pulses in one location is circled for clarity Pulse Repetition Rate of ~30 khz E E E E E E E E-03 Time (s) Figure 4-15: Pulse Repetition Rate of ~30 khz

36 Temperature (K) Temperature (K) Pulse Repetition Rate of ~40 khz E E E E E E E E-03 Time (s) Figure 4-16: Pulse Repetition Rate of ~40 khz Pulse Repetition Rate of ~50 khz E E E E E E E E-03 Time (s) Figure 4-17: Pulse Repetition Rate of ~50 khz

37 30 As can be seen, the pulse repetition rate of the laser has an effect on thermal fluctuations, specifically in the first two pulses. The temperature difference between the maximum surface temperature reached for the first and second pulses increases as pulse repetition rate is increased. This is because background temperature for the second pulse is higher since there is less time for heat to dissipate between pulses, leading to decreased laser energy deposition. Changing Heat Diffusion Coefficient The heat diffusion coefficient of the workpiece also has a significant effect on thermal fluctuations. The lower the heat diffusion coefficient, the slower heat is conducted throughout the material. When the heat diffusion coefficient is too low, heat conduction does not have an effect on thermal fluctuations because there is not enough time for heat to diffuse throughout the material; heat moves very slowly. When the heat diffusion coefficient is higher, heat moves very quickly and easily throughout the material. This causes a change in background temperature due to heat conduction and diffusion in the later repetitions of pulses (as described earlier in this chapter). However, when the heat diffusion coefficient is too high, heat is moving too quickly for any fluctuations to occur. While maintaining all other machining and material properties, the heat diffusion coefficient was varied from 0.05 to 0.1 to 0.2 centimeters squared per second. Four pulses were repeated in each location. Results found support the hypothesis. An example group of four pulses in one location is circled for clarity.

38 Temperature (K) Temperature (K) Heat Diffusion Coefficient of.05 centimeters 2 per second E E E E E E E E-03 Time (s) Figure 4-18: Heat Diffusion Coefficient of.05 centimeters squared per second Heat Diffusion Coefficient of.1 centimeters 2 per second E E E E E E E E-03 Time (s) Figure 4-19: Heat Diffusion Coefficient of.1 centimeters squared per second

39 Temperature (K) 32 Heat Diffusion Coefficient of.2 centimeters 2 per second E E E E E E E E-03 Time (s) Figure 4-20: Heat Diffusion Coefficient of.2 centimeters squared per second As can be seen, heat diffusion coefficient has an effect on thermal fluctuations. When the heat diffusion coefficient is too low, fluctuations due to heat conduction and dissipation are not prevalent because the heat does not move throughout the material quickly enough for an effect to be seen. Figure 4-18 shows maximum temperature reached at the surface of the material for four pulses repeated in the same location is decreasing for each pulse. When the heat diffusion coefficient is too high, fluctuations in the maximum temperature reached at the surface are also not prevalent because heat is diffusing away from the source point of the laser too quickly for a substantial increase in substrate temperature. Figure 4-20 shows nearly consistent maximum surface temperatures for four laser pulses. However, in Figure 4-19, fluctuations in maximum surface temperature are prevalent. These three results suggest an optimal range for heat diffusion coefficient to reduce thermal fluctuations.

40 Temperature (K) 33 Chapter 5 CONCLUSION In conclusion, background temperature affects energy absorption in ultra-short pulse laser nanomachining, and thus impact material removal. Background temperature is influenced by energy deposition from earlier pulses and heat conduction and diffusion. Higher background temperature at the surface implies lower energy deposition from a laser pulse. Consider the surface temperature of four laser pulses in the same location, graphed in Figured 5-1. Surface Temperature of 4 Pulses in Same Location E E E E-03 Time (s) Figure 5-1: Surface temperature of 4 laser pulses in the same location The simulation models four pulses of a laser in the same computational block. The increase in temperature for each pulse is dependent on the energy deposition for each pulse. This is dependent on the background temperature prior to each pulse.

41 Maximum Temperature (K) Background Temperature (K) To illustrate this point, the background temperature for the same four pulses is graphed in Figure Background Temperature E E E E-03 Time (s) Figure 5-2: Background temperature of 4 laser pulses in the same location Maximum surface temperature of the material for the same four pulses is graphed in Figure Maximum Temperature E E E E-03 Time (s) Figure 5-3: Maximum temperature of 4 laser pulses in the same location