Modelling thermal energy transfer in a femtosecond pulsed laser ablation of metal using a coupled spring-mass oscillator

Femtosecond pulsed laser ablation (fs-PLA) is an interesting yet complicated field of study especially for undergraduate students entering the field. Hence, a bridging concept using classical and mechanical analog will be helpful. In this paper, we modelled the thermal energy transfer between electron and lattice system in a fs-PLA of metal described by two temperature model (TTM) using a coupled spring-mass oscillator. This was achieved by providing correspondence of TTM parameters to the coupled spring-mass oscillator, with temperature as position, electron thermal conductivity as coefficient of friction, electron-phonon coupling factor as spring term, electron/lattice heat capacity as the mass m 1/m 2 respectively, and laser source term as the driving force. The thermophysical properties considered are temperature dependent leading to position dependent parameters of coupled spring-mass oscillator. Results showed that the coupled spring-mass oscillator exhibit many behavior similar to the TTM. Additionally, maximum positions achieved by m 2 behave similarly with maximum lattice temperature after achieving certain threshold value. However, many features of TTM such as spatial dependence and crater formation are not observed in the coupled spring-mass oscillator. Despite its limitation, the coupled spring-mass oscillator model was able to represent many features of the thermal energy transfer of fs-PLA, and could be an easy and useful model in understanding fs-PLA.


Introduction
The femtosecond pulsed laser ablation (fs-PLA) is a process that removes a portion of the material from the solid surface of the solid target by a focused femtosecond pulsed laser beam [1][2][3][4].Due to the controllable nature of the method, fs-PLA is used in different fields, including medicine [5], additive manufacturing [6], spectroscopy [7], among others.
Despite its experimental simplicity, the process behind fs-PLA is complicated covering various optical, mechanical, and thermal processes.In case of metallic target, one of the simplest and most popular approach in describing thermal energy transfer in fs-PLA is through two-temperature model (TTM).TTM describes fs-PLA via thermal energy transfer from electron system to lattice system [8][9][10] and is based on the heat diffusion within the metal after absorbing energy from the fs laser pulse [11,12].TTM is sufficient enough to describe the crater formed in the ablation process [8,13,14].Ablation is usually achieved when the combined energy of electron and lattice systems exceeds the energy needed for the metal to vaporize [10,15].This model considers both the spatial and temporal profiles of the electron and lattice temperatures which can provide insight into the dynamics of the laser heating process.Various parameters can be investigated and the effect of each can be simulated.
Mathematically, the TTM is described by the following equations: where C is the heat capacity, G is the electron-phonon coupling factor, T is the temperature, k is the thermal conductivity, and S is the laser source term.The subscripts e and l corresponds to electron and lattice respectively [8,9,16].Proper implementation of TTM requires the use of accurate optical and thermal properties of the material [8,[17][18][19].For example, the optical properties are governed by the dielectric function described by the Drudecritical point model [8,13,20].Thermal properties have complicated dependence in temperature.Electron thermal conductivity are dependent to electron and lattice temperature [20,21], while electron heat capacity and electron-phonon coupling factor are dependent to electron density of states and Fermi-Dirac distributions [10,17].These topics are complicated for undergraduate students beginning to do research in fs-PLA because they are usually encountered in advanced undergraduate or graduate-level courses.In many cases, these concepts are introduced with simplified assumptions, such as the temperature independence of the dielectric function, linear or T 3 dependence of the electron heat capacity, etc.However, these assumptions are rarely used in recent fs-PLA works, where temperatures achieved are high.A bridging concept using familiar topics can then help students gain a better understanding of fs-PLA.One example of simple classical description of a complicated intense laser-matter interaction (higher order harmonic generation) is the three step model (tunnel ionization, acceleration, and harmonic emission) [22,23].
Coupled spring-mass oscillators are common physical model used in describing many physical phenomena.Example includes the description of heat capacity using Einstein's model [24].Coupled spring-mass oscillators are spring-mass system where two or more masses connected via a spring and allows for an exchange of energy between the masses.Generally, we can describe the motion of the system into four stages: (1) one mass will experience driving force leading to its acceleration, (2) the spring placed between the masses will allow the force to be applied on the other mass, (3) subsequently, there will be a motion dictated by the condition of the oscillator, (4) until the two bodies arrive at equilibrium motion.Coupled spring-mass oscillators are regularly encountered in many undergraduate treatise in classical mechanics [25,26], making it an ideal starting point for a mechanical analog of a physical process.
In this paper, we will model the thermal energy transfer in fs-PLA of metal described by TTM using a coupled spring-mass oscillator model.We will represent the thermophysical properties of the electron and lattice system using variables of coupled spring-mass oscillator with temperature as position, thermal conductivity as coefficient of friction, electron-phonon coupling factor as spring term, electron and lattice heat capacity as mass m 1 and m 2 respectively, and laser source term as the driving force.The parameter correspondence from TTM to coupled spring-mass oscillator are summarized in table 1. Analogous temperature temporal profile of the electron and lattice system is generated using coupled spring-mass oscillator model.We considered the case where thermophysical properties are temperature dependent to accurately represent the temperatures achieved in fs-laser processing.Similar determination of threshold fluence from TTM are implemented using coupled spring-mass system.We have used copper as our metal to demonstrate our work.

Theoretical framework 2.1. Two temperature model
The electron T e ( ) and lattice temperature T l ( ) of the copper under the fs-PLA are related to each other by TTM as described in equations (1a) and (1b).The highly forwarded energy from the laser bean limits the transfer of energy towards the material (along z-axis).The temperature derivative along the x − and y − axis is negligible since the lateral transfer of energy in the surface is minimal.
The electron heat capacity (C e ) can be analytically determined by using equation (2).The equation includes the Fermi-Dirac distribution ( f (ò, μ chem , T e )) electron density of state (g(ò) at a particular energy level ò [17,27].
Similar to C e , the electron-phonon coupling factor (G) determined by Equation 3 also depends on Fermi-Dirac distribution the electron density of state [17,28].
To facilitate faster computation, we employed 10th order polynomial fitting function to the otherwise complicated analytical integration of equations (2) and (3) in our TTM calculations.Figures 1(a) and (b) shows the electron heat capacity and electron phonon coupling factor as function of electron temperature [17].The lattice heat capacity of the copper metal used in the calculation is 3.46 × 10 6 Jm −3 K −1 [11].
The electron thermal conductivity of copper used used in this work is given by the Anisimov model [ where T T l l F J = and T T e e F J = , with T F the Fermi temperature of the metal.For copper, the quantities T F , χ, and η are equal to 8.16 × 10 4 K, 377Wm −1 K −1 , and 0.139 respectively [20,21]. Figure 1(c) shows the electron thermal conductivity obtained with T l = 3000 K.The lattice thermal conductivity, is small compared to the electron thermal conductivity and is taken to be 1% of the k eq of copper [21].
where t p is the laser pulse duration, |R| 2 the reflectivity of the copper, J the laser fluence, ω 0 is the laser beam waist, d is the the optical penetration depth and λ B is the ballistic length of copper.In this work, we used d = 13 nm, λ B = 15 nm, and t p = 10 fs, .

Coupled spring-mass oscillator
The coupled spring-mass oscillator model that we used in this study is made by connecting two masses using a spring.The coupled spring-mass oscillator is placed on the surface, where the first mass is being pushed by a driving force F d (t) to the right.To implement the position dependence of the coupled spring-mass parameters (corresponding to temperature dependent parameters in TTM) we introduce the following modifications: the surface encountered by m 1 has position dependent coefficient of friction, the m 1 is represented as a box that is being filled with sand (via tube above m 1 ) as it moves, and the massless spring is represented by a combination of co-axial springs with different length and number of coils.Figure 2 illustrates the coupled spring-mass oscillator used in this study.m 1 will experience a force from driving force F d (t), frictional force F f1 , and spring force F s1 , while m 2 is affected by the frictional force F f2 , and spring force F s2 .The position dependence of m 1 is obtained from scaling with polynomial function based on the temperature dependence of the electron heat capacity given by equation (2) [17,29] and is shown in figure 1(d).The contact of the two masses on the surface provides the frictional force given by: where g is gravitational constant, μ 1 (x 1 ) is the position dependent coefficient of friction on the surface on m 1 , and μ 2 the position independent coefficient of friction on the surface on m 2 .μ 1 (x 1 ) was scaled with polynomial function based on the electron thermal conductivity as described by equation 4 [21,29] and is shown in figure 1(f).We also considered the direction and shift of static to kinetic friction of the two masses [30,31].These are incorporated into the model using the following equation: where a = − 2, b = 3, c = 1.4,x 1 ¢ is the velocity of m 1 and x 2 ¢ is the velocity of m 2 .The coefficient f 1 and f 2 changes the sign of F f1 and F f2 depending on the velocity.The spring is compressed if the displacement of m 1 is greater than m 2 , otherwise the spring is stretched.We define the spring force acted on the two masses by the equation where k is the position dependent spring term obtained by scaling with polynomial function based on the temperature dependent of electron-phonon coupling factor given by equation (3 [17,29], and is shown in The equations of motion of the two masses are given by We used a Gaussian profile driving force with amplitude M, standard deviation of t 2 s , and peak at t 0 to represent the laser source term S. We used the value of t 0 = 10 s with t 10 s s = in this work.We assign m 2 to be 3.46 kg. We solved the coupled differential equation on equation (9) using numerical methods implemented in Wolfram Mathematica.We obtained the position-time plot of m 1 and m 2 , representing the temperature profile of electron and lattice system respectively.We also record the maximum position of m 2 to investigate the behavior of the coupled spring-mass oscillator with the driving force.

Results and discussion
In general, the thermal energy transfer in fs-PLA as described by TTM have four stages: (1) absorption of laser energy by the electron system resulting to an initial increase in electron temperature, (2) transfer of the thermal energy from the electron system to the lattice system via electron-phonon coupling, (3) increase the lattice temperature, and (4) achieving equilibrium between electron and lattice system [1,8].In the context of the coupled spring-mass oscillator, this corresponds to: (1) acceleration of m 1 during the application of driving force, (2) transfer of mechanical energy from m 1 to m 2 due to the coupling enabled by spring, (3) acceleration of m 2 , and (4) achieving equilibrium position of m 1 and m 2 .Figure 3

summarizes this process
These are all seen in figure 4 where we simulate both the (a) TTM and (b) spring-mass oscillator model with parameters provided in section 2. We can see in (a) that the electron temperature instantaneously increased upon absorbing the laser energy.T e ( ) slowly decrease as it transfers the energy to the lattice system making the temperature of lattice slowly rise.Eventually, equilibrium temperature of the two systems is achieved.Correspondingly in (b), the position of m 1 started to increase when the driving force is applied, and m 2 followed eventually.This signifies the energy transfer from m 1 to m 2 .
We can also see that increasing the laser fluence in (a) led to an increased electron and lattice temperatures.Likewise, when we increase the magnitude of the driving force in (b), the position of m 1 and m 2 also increased.
There are some differences despite many similarities in the results of TTM and coupled spring-mass oscillator.The first includes the relative magnitude of the position as compared to the temperature.Maximum electron temperature is higher than maximum lattice temperature.In contrast, maximum position of m 2 is higher than m 1 .Another difference is the respective temperature profile of position and temperature.m 1 immediately achieved a stable position, comparable to the behavior of lattice temperature and not the electron temperature.These observations are due to the combination of increased mass of m 1 and higher frictional force on m 1 , allowing it to move in a shorter distance and stopping at faster time.Nevertheless, important features of the transfer of thermal energy in TTM are qualitatively observed in the proposed coupled spring-mass oscillator model.Figures 5(a) and (b) shows the maximum lattice temperature at increasing fluence, and maximum position of m 2 at increasing driving force.Both TTM and coupled spring-mass oscillator showed a similar behaviour when a catalyst is applied in the system (fluence for TTM and driving force for coupled spring-mass oscillator).The maximum temperature and position showed a stagnant value until a certain threshold where it abruptly increase.In the context of TTM, the threshold determination is done using the ablation rate (or crater depth / number of pulses) [32,33].However, the correspondence of ablation rate in the context of coupled spring-mass oscillator can be complicated.In its simplified form, ablation of metal will occur if the lattice temperature exceeds the vaporization temperature.In the coupled spring-mass oscillator, this should correspond to the position of m 2 surpassing an arbitrary "vaporization "distance.Nevertheless, the coupled spring-mass oscillator captures the logarithmic dependence in fs-PLA and is proved to be an effective approach in obtaining a mechanical analog of a threshold.Some features of the TTM are not seen in the proposed coupled spring-mass oscillator.The first is the absence of the correspondence of spatial profile of the results of TTM in the coupled spring-mass oscillator.This spatial profile allows the determination of crater dimension using TTM.Because of this, we are not able to describe the crater formed in the context of coupled spring-mass oscillator.Despite the sophistication of the proposed coupled spring-mass oscillator model, it has its limitations including the mechanical means to decrease the mass for decreasing position.This prevents the researchers to explore the reversibility of the process (changes with decreasing temperature).Nevertheless, the proposed coupled spring-mass oscillator model of thermal energy transfer in fs-PLA provides many mechanical analog of TTM.This should help undergraduate students to appreciate faster the complex field of pulsed laser ablation.

Conclusion and recommendations
We proposed that the thermal energy transfer in femtosecond pulsed laser ablation as described by the two temperature model can be modelled using coupled spring-mass oscillator.This was achieved by providing correspondence of two temperature model parameters to the coupled spring-mass oscillator.The proposed oscillator took into consideration the temperature dependence of the two temperature model parameters.Features of the two temperature model observed in the oscillator includes transfer of energy from electron system to lattice system (transfer of energy from m 1 to m 2 ), initial increase in electron temperature before lattice temperature (increase in position of m 1 before m 2 ), tendency to achieve equilibrium (achieving the same position) after some time, and an abrupt increase in the lattice temperature (position of m 2 ) after reaching some threshold value.We enumerated some limitations of the coupled spring-mass oscillator in modelling thermal energy transfer in fs-PLA including the absence of ablation rate and ablation depth.We suggest taking into account the elastic limit of the spring into the spring-mass oscillator equations to get the maximum deformation allowed, and therefore maximum position allowed for m 2 before the spring would permanently deformed, much the same way a material is permanently ablated from the surface.Despite the absences of some features of two temperature model, we conclude that the coupled spring-mass oscillator is a good introductory approach for students in understanding the complex topic of femtosecond pulsed laser ablation.

Figure 1 .
Figure 1.Two-temperature model parameters: (a) Heat capacity, (b) electron-phonon coupling factor, and (c) electron thermal conductivity (obtained with T l = 3000 K) as function of electron temperature.The corresponding coupled spring-mass oscillator parameters are: (d) mass m 1 , (e) spring term k, and (f) coefficient of friction μ 1 , as function of position/displacement.

Figure 2 .
Figure 2. Schematic diagram of the coupled spring-mass oscillator model.m 2 is a solid box on top of a uniform surface, while m 1 is a box that accommodates sand (via tube above m 1 ) as it moves.m 1 experiences position dependent friction and spring force.The circled numbers corresponds to the order of energy transfer.

Figure 3 .
Figure 3. Energy transfer process in TTM and system of motion in a spring-mass oscillator.The flowchart shows the analogous stages of the two models.

Figure 4 .
Figure 4. (a) Temperature vs time plot of the electron (T e -solid line) and lattice system (T l -dashed line) at difference fluence obtained using TTM.(b) Position vs time plot of the m 1 (solid line) and m 2 (dashed line) at different driving force obtained using the proposed coupled spring-mass oscillator.

Figure 5 .
Figure 5. (a) lattice temperature vs fluence plot from TTM.(b) Maximum position vs driving force amplitude plot from coupled spring-mass oscillator.

Table 1 .
TTM parameters with their corresponding coupled springmass oscillator parameter.