Abstract
Nowadays, lasers are used in a wide variety of manufacturing processes, such as cutting, sintering and welding. The evolution of laser technologies has led to the creation of ultrashort pulsed lasers, with a pulse duration below 10 ps, which have the ability, on the contrary with the conventional lasers, to stimulate separately the electrons and the lattice. Thus, two different temperatures, the electron temperature and the lattice temperature appear. This cannot be described by the classical Fourier heat equations and thus the Two-Temperature Model (TTM) has been proposed. In the TTM, a heat equation that describes the electron temperature is conjugated with a heat equation that describes the lattice temperature. Except from the correct implementation of the heat equations, other factors that should be taken into consideration during the development of the TTM simulation are the correct and accurate modelling of the ultrashort pulsed laser and the appropriate selection of the computational method regarding the targets of each specific study. The aim of this review paper is not only to present the current literature regarding the different TTMs, ultrashort pulsed laser models and computational methods, but also to create mind maps that will help the researcher to choose the most appropriate TTM and computational method regarding the targets of each specific study. Moreover, in this review paper, recommendations for future work are given, regarding the more accurate and realistic modelling of the laser source.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Laser Beam Machining (LBM) is a hot topic, nowadays, into the Mechanical Engineering world, as it includes many advanced manufacturing processes, such as laser cutting, laser drilling, laser marking, laser welding, laser sintering and laser heat treatment. The implementation of lasers into manufacturing processes is crucial, as strict design requirements and complex shapes can be achieved for a wide variety of materials, which is not so easily achievable with conventional machining [1]. A laser, which is the acronym for “Light Amplification by Stimulated Emission of Radiation”, is a device in which stimulated emission of electromagnetic radiation takes place, which leads to optical amplification and eventually to the emission of light. Lasers’ most important property is their ability to emit spatially and temporally coherent light. Spatial coherence allows laser beams to stay focused to a tight spot, whereas temporal coherence allows lasers to stay narrow over great distances [2]. Lasers are complex devices which consist of many different parts, such as the energy pumping source, the gain medium, which is included into the optical cavity, the optical cavity, which includes two mirrors in between of which the photons are reflected till they achieve the desired intensity and the output coupler (Fig. 1).
A basic laser device
The laser beam formation is based on the stimulated emission phenomenon. Specifically, the energy pumping source supplies the gain medium with energy. This energy is absorbed by the gain medium particles and thus they jump from their ground state to an excited state, which is in a higher energy level. During the return of each particle into its ground state, a photon is emitted, which then stimulates another particle. Gradually, by this process, the number of particles at excited state becomes bigger than the number of particles at ground state and thus population inversion is achieved. With population inversion, when the laser beam penetrates the gain medium, it has a much higher possibility to lead to stimulated photons emission than to stimulated photons absorption and thus beam amplification is achieved. When the laser beam comes out of the output coupler to the environment, its light intensity has a Gaussian distribution, which is perpendicular to the direction of the laser beam [3].
Lasers are operating in either continuous or pulsed mode. In Continuous-Wave (CW) lasers, their beam output power is constant over time, whereas in pulsed lasers, the output power appears as pulses of some duration that are repeated periodically. Pulse durations may range from some milliseconds (conventional pulsed lasers) to some nanoseconds (short pulsed lasers) down to some picoseconds or femtoseconds (ultrashort pulsed lasers) [4]. From all these laser types, the current review paper focuses on the modeling of the ultrashort laser irradiation on metallic surfaces. In order to understand the importance of ultrashort lasers, they are compared to the conventional and short pulsed lasers. In conventional and short pulsed lasers, laser pulse duration is higher than both the electron and the lattice relaxation times, so both electrons and the lattice are stimulated together by each laser pulse. On the other hand, at ultrashort lasers pulse duration is higher than the electron relaxation time, but lower than the lattice relaxation time. Thus, in ultrashort lasers, only electrons are stimulated by the laser pulse and increase their temperature. After a temporal delay and due to the interactions of the stimulated electrons with the lattice, the lattice particles are also stimulated and start to increase their temperature. For this reason, in ultrashort laser irradiation, two different temperatures are appeared, the electron temperature and the lattice temperature [5]. For this reason, the classical Fourier heat transfer models are not appropriate to simulate the irradiation of ultrashort lasers. So, in order to simulate the irradiation of the ultrashort lasers, the Two-Temperature Models (TTMs) have been proposed in the literature. According to the simplest form of TTM (parabolic TTM), two different Fourier heat equations are used. The one describes the electron temperature and the other describes the lattice temperature. These two equations are conjugated by an electron–phonon coupling factor [6]. However, more complex, but also more accurate TTMS have been proposed that include the electron relaxation time (hyperbolic TTM), the lattice relaxation time (dual hyperbolic TTM), the electron drift velocity (Semiclassical TTM) and only macroscopical parameters (dual-phase lag model) [7].
Furthermore, a very important factor that should be taken into consideration when building the TTM simulations is the modeling of the ultrashort pulsed laser source. Pulsed lasers are described mathematically via spatial and temporal distributions. The laser spot is, usually, described by a Gaussian distribution regarding the x and y-axis and by a uniform distribution that follows the Beer-Lambert law regarding the z-axis. On the other hand, for the temporal description of the laser irradiation, a variety of distributions have been proposed, such as the rectangular distribution, the Gaussian distribution, the hyperbolic secant distribution and the Lorentzian distribution [8].
Finally, a significant parameter for all the simulations is the computational method used. In literature, analytical methods, Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), Meshless Methods (MM), Molecular Dynamics (MD) and Artificial Neural Networks (ANN) have been used to solve the TTM equations and estimate the electron and lattice temperatures [9]. All these computational methods have their advantages and disadvantages regarding the targets of each study.
The aims of the current review paper are the following:
-
Presentation, analysis and comparison of all the basic TTMs appeared in the literature (parabolic TTM, hyperbolic TTM, dual hyperbolic TTM, semiclassical TTM, dual-phase lag model) that describe the irradiation of metallic surfaces by ultrashort pulsed lasers.
-
Presentation, analysis and comparison of a variety of ultrashort pulsed laser models (rectangular distribution, Gaussian distribution, hyperbolic secant distribution, Lorentzian distribution) that have been appeared into both the Engineering and Physics literature.
-
Presentation, analysis and comparison of a variety of computational methods (analytical methods, FDM, FEM, FVM, MM, MD, ANN) that have been used in the literature to solve the TTM equations.
However, the aims of this review paper are not limited only to a simple presentation of the models (TTMs and laser sources models) and the computational methods, used to solve the TTM equations, but they are also extended to the novel paths of a critical assessment. Specifically, the novel aims of this review paper are the following:
-
Creation of a mind map that will help the researcher choose which TTM is more appropriate regarding the parameters of the TTM problem and the targets of each specific study.
-
Recommendations for future work regarding the modeling of the ultrashort pulsed lasers.
-
Creation of a mind map that will help the researcher choose the most appropriate computational method regarding the targets of each specific study.
2 Theory of Ultrashort Pulsed Laser Irradiation of Metallic Surfaces
2.1 Parabolic One Step Model (POS)
In classical heat transfer theory, Fourier law is a fundamental equation, which describes the heat conduction flow [10]:
where \(q\) is the heat flux vector, \(k\) is the thermal conductivity and \(T\) is the temperature.
If this equation is combined with the heat source input \(S\) and the general form of the energy equation:
where \(\rho and {C}_{p}\) are the material density and heat capacity, respectively, then the classical Fourier heat conduction equation (parabolic one step model- POS) accrues:
In laser machining, this equation is widely used for describing heat transfer into the irradiated material. In this case, \(S\) stands for power per volume induced to the material due to the absorption of the laser irradiation. However, for reasons that will be analyzed into the next paragraphs, the classical Fourier heat conduction equation can only describe accurately the irradiation of conventional and short pulsed lasers (for example millisecond and nanosecond pulsed lasers). For ultrashort lasers, more advanced equations are needed.
Lasers are considered as ultrashort when their pulse duration is shorter than all the metal`s lattice relaxation processes, such as the electron-to-lattice energy exchange, the hydrodynamic motion and the heat diffusion. Thus, during the laser pulse and after that till the electron-lattice interaction time is reached, the atomic structure of the metal remains undisturbed [11]. As a practical rule, in bibliography [6], it is considered that lasers with a pulse duration shorter than 10 ps are ultrashort. On the other hand, the electron relaxation time is much shorter than the laser pulse duration, so when the ultrashort laser irradiates matter, the electrons are excited. For this reason, the temperature of the electrons increases drastically, whereas the lattice remains cold. This huge difference between the electrons and the lattice temperatures leads to non-equilibrium phase change conditions, that cannot be described accurately by the conventional Fourier law for heat conduction [6, 11].
To sum up, when a metal is irradiated by an ultrashort laser, three main physical processes take place [12]:
-
(a)
Firstly, free electrons, within the optical penetration depth, are excited due to the absorption of the laser energy. Electron heat capacity is very low, thus the electron temperature follows the shape of the laser pulse without delay [13]. Afterwards, two competing processes take place: a) the ballistic motion of the excited electrons deeper into the sample, with near to Fermi velocities and b) collisions between the excited electrons and the electrons around the Fermi level. The s/p-band metals seem to be affected more by the ballistic motion, whereas d-band metals seem to be affected more by the electron collisions [14]. Thus, this first step is determined by the electrons relaxation time, which is the average time between two consecutive electron collisions [15]. For this reason, the electrons temperature increases drastically, whereas the lattice remains cold.
-
(b)
Secondly, energy is diffused to the lattice, because of the electron–phonon collisions. As mentioned previously, the lattice is not heated at the same time with the electrons. The ion mass is much bigger than the electron mass, so the electron–ion collisions cannot transfer much energy to the lattice [13]. Due to electrons` very low mass, the electron Fermi velocity is greater than the phase velocity of light in the lattice. This is a Cherenkov radiation case. Specifically, as it is well known from classical physics, accelerating excited particles (in this case, electrons) emit electromagnetic waves, which form spherical wavefronts, propagated with the phase velocity of the medium, according to the Huygens principle. Thus, when any excited particle passes through a medium, it polarizes the medium`s particles (in this case, ions) around it. Thus, the medium`s particles get excited. When they return to their ground state, they re-emit the “excitation” energy as photons. These photons are responsible for the formation of the spherical wavefronts [16]. The ions` vibrations, due to the medium`s particles excitation, are treated as phonons, according to the Debye theory. Thus, the lattice heat capacity depends on temperature [17]. This stage is determined by the electron-lattice interaction time, which is the average time between two consecutive polarizations of an ion from the electrons.
-
(c)
Finally, phonon–phonon heat transfer leads to energy diffusion in the lattice. Here, the heat transfer occurs due to the ion-ion collisions. In this stage, a very important parameter is the thermalization time, which is the time when thermal equilibrium between the electrons and the lattice is reached.
2.2 Parabolic Two Step Model (PTS)
In order to describe mathematically the above phenomena, firstly Anisimov and Rethfeld [13] introduced the Two-Temperature Model (TTM). The kinetics of electron-lattice relaxation, given by the equations [18, 19]:
(where \(c\,and\,G\) are the specific heats and the electron–phonon coupling factor and \(e\,and\,i\) stand for the electron and lattice subsystems respectively), were combined with the classical Fourier heat conduction Eq. (3). Thus, the following two coupled equations accrue:
where \(S\), now is induced, due to the absorption of the laser irradiation, only to the electrons and not to the whole material.
Based on estimations, Anisimov and Rethfeld excluded the phonon heat conduction from (5), as its contribution could be considered negligible for most of the cases. Moreover, because (5) is a system of parabolic equations, this model is called Parabolic Two Step Model (PTS).
For (5), it is assumed that the electron and phonon heat transfer can be described by the Fourier law. This is valid only when the characteristic space and time scales of the temperature field are much greater compared to the electrons and ions` mean free path and relaxation time, respectively [13]. This implies that the electron and lattice subsystems are in local equilibrium with themselves and the energy in each subsystem is transported only diffusively. However, for much shorter laser pulses, these equations are not applicable, mainly for two reasons. Firstly, according to (5) energy is transported with infinity speed, which is not true according to Joseph and Preziosi [20], who mentioned a finite speed of energy propagation. Secondly, it is observed that the energy in each one of the electron and lattice subsystems is transported not only diffusively but also ballistically [21]. When a metal is irradiated by an ultrashort laser, the free electrons (which are the excited electrons that left their corresponding atoms and lie close to the Fermi surface) move with very high speed, close to their Fermi velocity and propagate through the metal without experiencing any scattering. This collisionless heat transport between the electrons is called ballistic phenomenon. On the other hand, the rest electrons (those which were not excited by the laser irradiation and are close to their corresponding atom) move with much less speed, so the heat transfer between these electrons occurs only via collisions. This kind of heat transport is called diffusive phenomenon [11].
2.3 Hyperbolic One Step Model (HOS)
In order to include the phenomena above, non-Fourier equations should be used. The first trial came from Maurer [22] who applied the Cattaneo-Vernotte equations [23, 24] to the heat conduction of metals. Practically, Mauer generalized the Fourier law by introducing the relaxation time at the heat flux-temperature gradient dependency [25]:
which can be written, according to the Taylor series expansion [26] as:
Thus, Maurer’s model (hyperbolic one step model- HOS) is the following:
where \({\tau }_{e}\) is the electron relaxation time and \({k}_{eq}\) is the thermal conductivity, assuming that phonons are in equilibrium with themselves all the time. This hyperbolic one step heat conduction model is based on two assumptions: (a) Interactions between free electrons-electrons are negligible compared to free electron–phonon interactions and (b) The phonons are always in thermal equilibrium with themselves, thus their distribution functions are unaffected by the free electron–phonon interactions.
2.4 Hyperbolic Two Step Model (HTS)
Because Maurer’s model does not consider the energy exchange between the electrons and the lattice, it cannot be used to describe the ultrashort pulse laser heating of metals. In order to include electron–phonon interactions, Qiu and Tien [15] derived the following hyperbolic two step model (HTS) based on the Boltzmann transport equation for electrons:
Although this model can accurately describe the electron temperature throughout the process, it is not valid to describe and the lattice temperature. Specifically, it can only predict correctly the lattice temperature at the starting stages, but lacks validity as the process progresses, because the lattice heat conduction is excluded from the model.
2.5 Dual Hyperbolic Two Step Model (DHTS)
To solve the limitation of the HTS model, Chen and Beraun [27] generalized the previous model in order to include and the lattice heat conduction, as follows:
This model is known as Dual Hyperbolic Two Step Model (DHTS).
2.6 Semiclassical Two Step Model (STS)
However, Chen et al. [28] observed that for very high-intensity and short-pulse lasers, the contribution of electron drift velocity to thermal transport is comparable with the contribution of the diffusion. The electron drift velocity is the result of the electric field and the hot-electron kinetic pressure. In order to include the electron drift velocity, the following Semiclassical Two Temperature Model is introduced:
where \(m\) is the mass, \(v\) is the drift velocity, \(e\) is the charge of an electron, \({\mu }_{0}\) is the mobility of electrons at room temperature and \(\beta\) is the thermopower, which equals to [29]:
where \({T}_{F}\) is the Fermi temperature.
2.7 Estimation of the Electron Properties and the Optical Properties
In order to implement all the above equations, firstly the temperature dependent properties of the electrons and the optical properties should be calculated. Specifically, the electron-lattice coupling factor G is given by [30]:
where \({n}_{e}\) is the free electron density and \({c}_{s}\) is the bulk material speed of sound estimated by:
where \(B\) represents the bulk modulus and \({\rho }_{m}\) the bulk material density.
Electron heat capacity is given by [31]:
where \(V\) represents the volume and \(\langle \varepsilon \rangle\) is the average kinetic energy per electron, which is estimated by:
where \(\varepsilon\) shows a free electron kinetic energy, \({k}_{B}\) is the Boltzmann constant, \({\rho }_{s}\) is the states density, which is calculated by:
where \(h\) stands for the Planck constant.
Moreover, the chemical potential \(\mu\) is given by [29]:
where \({e}_{F}\) is the Fermi energy and is estimated by:
If \(0<{T}_{e}<0.1{T}_{F}\), \({c}_{e}\) equation is simplified to [31]:
where \(\gamma\) is the electron heat capacity constant.
Furthermore, free electron heat conductivity is calculated according to the Drude theory [29]:
where \({v}^{2}\) shows the mean square speed.
Moreover, electron relaxation time \({\tau }_{e}\) is calculated by [32, 33]:
where \({Z}^{*}\) represents the ionization state, \(e\) shows the electron charge, \({F}_{1/2}\) is the Fermi integral and \(\mathrm{ln}\Lambda\) is the Coulomb logarithm given by [34]:
where the maximum collision parameter \({b}_{max}\) is calculated by:
and the minimum collision parameter bmin is given by:
where ω is the laser angular frequency and ωp is the plasma angular frequency estimated by:
where \({\varepsilon }_{0}\) is the free space electrical permittivity.
2.8 Dual Phase Lag Model (DPL)
As it can be observed, all the TTMs (HTS (9), DHTS (10), STS (11)), except from the PTS (5), are based on the single-phase lag model, introduced by Tzou [35]. The general single-phase lag equation is the following:
where \(\tau\) is the heat flux lag period due to thermal inertia.
Although all the variations of single-phase lag model solve the problem of infinity speed of energy propagation of the PTS, still they have some drawbacks or abnormalities. Bai and Lavine [36] showed that hyperbolic models may give temperatures below absolute zero for some conditions, whereas Zhang’s [37] hyperbolic models predicted the generation of negative entropy. Moreover, in order to solve each single-phase lag model, it is necessary to estimate the values of (13)-(26). To calculate these values, knowledge of the microscopic properties of the electrons and the lattice atoms is needed. These microscopic properties are sometimes very difficult to be calculated and they also maybe out of the interest of the engineers. So, there is the need to correlate these microscopic properties with the macroscopic heat transfer properties.
In order to cope with all the above abnormalities and drawbacks, Tzou [38] proposed the Dual Phase Lag model (DPL). In this model, in order to disappear the abnormalities, a second lagging period is introduced, as it is shown into the following general equation, which represents the lagging behavior:
where \({\tau }_{q}\) shows the heat flux phase lag due to thermal inertia (spatial effect of heat transfer) and \({\tau }_{T}\) shows the temperature gradient phase lag due to the electron-lattice interaction (temporal heat transfer).
A Taylor expansion of the above equation gives:
If this equation is combined with the general form of the energy Eq. (2) and the heat flux vector is eliminated, then the temperature representation of the dual-phase lag model is given:
or in terms of thermal diffusivity \(a\):
As it is observed, with DPL models all the heat transfer phenomena can be described by only a one-temperature formulation, the material temperature. Furthermore, now the micro-scaling thermal diffusivity, electron–phonon interaction and phonon scattering models can be described by the macro-scaling \({\tau }_{T}\) and \({\tau }_{q}\) [39]. Moreover, according to the thermal wave theory, which is the basis of all the phase lag models, relaxation behavior is the fundamental mechanism that bridges thermal diffusivity (micro-scale property) to thermal wave speed (macro-scale property), regarding to [35]:
where \(C\) is the material thermal wave speed.
Other derivations from the thermal wave theory are the following [20]:
So, by combining (32), (33) and (34), the relations that combine the macroscopic \({\tau }_{T}\) and \({\tau }_{q}\) with the microscopic properties can also obtained:
3 Modeling of the Ultrashort Pulsed Laser Heat Source
Pulsed lasers can be described mathematically via spatial and temporal distributions.
3.1 Spatial Distribution of the Laser Spot According to x and y-Axis
Laser spot can be described by a Gaussian distribution according to x and y axis. The intensity of this distribution according x-axis, \({I}_{x\,spatial}^{gaussian}\) is given by [40, 41]:
where \({r}_{o}\) is the laser beam radius, assuming that it is estimated at position 1/e.
The probability density \({S}_{x\,spatial}^{gaussian}\) of the above equation is of the following form:
where \({A}_{x\,spatial}^{gaussian}\) is the coefficient used to normalize \({S}_{x\,spatial}^{gaussian}\) to 1 and is calculated by:
So,
According the y-axis, the same procedure is followed and eventually the probability density according y-axis \({S}_{y\,spatial}^{gaussian}\left[\frac{1}{m}\right]\) is:
Combining x and y-axis, the probability density according x and y-axis, \({S}_{x,y\,spatial}^{gaussian}\) is [42, 43]:
3.2 Spatial Distribution of Laser Irradiation According to z-Axis
Spatial irradiation according to z-axis follows the Beer-Lambert law and the ballistic range. Given that the absorbed laser energy is spread deeper into the sample due to both the ballistic motion and the hot electron diffusion, Hohlfeld et al. [14] suggested to incorporate the ballistic range \({\delta }_{b}\) together with the optical penetration depth \({\delta }_{s}\).
Usually, the spatial irradiation according to z-axis is described by a uniform distribution, with intensity \({I}_{z\,spatial}^{uniform}\) [12]:
where \({\delta }_{s}\) is the optical penetration depth and \({\delta }_{b}\) is the ballistic range of electrons.
The probability density \({S}_{z\,spatial}^{uniform}\) of the above equation is of the following form:
where \({A}_{z\,spatial}^{uniform}\left[\frac{1}{m}\right]\) is calculated by:
where \(L\) is the film thickness.
So [27],
3.3 Temporal Distribution of Laser Irradiation
Temporal distribution of laser irradiation, usually has a mean value of \(2{t}_{p}\) and a temporal period of \(4{t}_{p}\), so for the 1st period [44]:
where \({t}_{p}\) is the laser pulse duration, given as the full width at half maximum laser pulse intensity.
A variety of distributions have been proposed to describe the temporal irradiation of a laser. The most common distributions are the rectangular distribution, the Gaussian distribution, the hyperbolic secant distribution and the Lorentzian distribution.
3.3.1 Rectangular Distribution
The intensity of a temporal rectangular distribution \({I}_{temporal}^{rectangular}\) is [44]:
The probability density \({S}_{temporal}^{rectangular}\) of the above equation is of the following form:
where \({A}_{temporal}^{rectangular}\) is calculated by:
Thus,
3.3.2 Gaussian Distribution
The intensity of a temporal gaussian distribution \({I}_{temporal}^{gaussian}\) is [41]:
The probability density \({S}_{temporal}^{gaussian}\) of the above equation is of the following form [45]:
where \({A}_{temporal}^{gaussian}\) is calculated by [46]:
So,
However, the Gaussian distribution is \(\propto {e}^{-{t}^{2}}\), so the wings of the distribution are falling quickly, thus this distribution is not always appropriate to describe real experiments [44].
3.3.3 Hyperbolic Secant Distribution
The intensity of a temporal hyperbolic secant distribution \({I}_{temporal}^{{sech}^{2}}\) is [44]:
The probability density \({S}_{temporal}^{{sech}^{2}}\) of the above equation is of the following form:
where \({A}_{temporal}^{{sech}^{2}}\) is calculated by [47]:
Assuming that \(k=\frac{2\mathit{ln}\left(1+\sqrt{2}\right)}{{t}_{p}} and u=k\left(t-2{t}_{p}\right)\):
Because,
and
Eventually,
So,
The hyperbolic secant distribution is \(\propto {e}^{-t}\), so the wings of the distribution are falling slowly, thus this distribution is often more appropriate to describe real experiments accurately.
3.3.4 Lorentzian Distribution
The intensity of a temporal Lorentzian distribution \({I}_{temporal}^{lorentzian}\) is [44]:
The probability density \({S}_{temporal}^{lorentzian}\) of the above equation is of the following form:
where \({A}_{temporal}^{lorentzian}\) is calculated by [8]:
So,
The Lorentzian distribution is \(\propto \frac{1}{t}\), so the wings of the distribution are very large, thus this distribution may also be useful to describe real experiments.
3.4 Full Spatial and Temporal Modeling of the Ultrashort Pulsed Laser
In order to estimate the full distribution over space and time, the x,y,z-spatial and temporal distributions above, should be combined. The combination of the above distributions, leads to four different cases:
3.4.1 Case 1: Rectangular temporal distribution
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and rectangular-temporal sub-distributions \({S}_{x,y,z\,spatial}^{rectangular\,temporal}\) is:
3.4.2 Case 2: Gaussian temporal distribution
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and Gaussian-temporal sub-distributions \({S}_{x,y,z\,spatial}^{gaussian\,temporal}\) is:
3.4.3 Case 3: Hyperbolic secant temporal distribution
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and hyperbolic secant-temporal sub-distributions \({S}_{x,y,z\,spatial}^{hyperbolic\,secant\,temporal}\) is:
3.4.4 Case 4: Lorentzian temporal distribution
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and Lorentzian-temporal sub-distributions \({S}_{x,y,z\,spatial}^{lorentzial\, temporal}\) is:
As it is observed, all the above distributions are normalized to 1. However, in order to model the heat source irradiation, it is necessary to normalize these distributions to the laser energy absorbed by the metal \((1-R)E\), where \(E\) is the total laser energy and \(R\) is the reflectivity of the metal. Thus, in order to normalize to \((1-R)E\), each one of the four above cases should be multiplied with \((1-R)E\). Moreover, it is observed [48] that \(\frac{E}{\pi {r}_{o}^{2}}\) equals the laser beam fluence \(F\), so \(F=\frac{E}{\pi {r}_{o}^{2}}\). Eventually, the four above cases are given as follows:
3.4.5 Case 1: Rectangular temporal distribution (normalized)
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and rectangular-temporal sub-distributions \({S}_{x,y,z\,spatial,normalized}^{rectangular\, temporal}\), which is normalized to \(\left(1-R\right)E\), is:
3.4.6 Case 2: Gaussian temporal distribution (normalized)
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and Gaussian-temporal sub-distributions \({S}_{x,y,z\,spatial,normalized}^{gaussian\, temporal}\), which is normalized to \((1-R)E\), is:
3.4.7 Case 3: Hyperbolic secant temporal distribution (normalized)
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and hyperbolic secant-temporal sub-distributions \({S}_{x,y,z\,spatial,normalized}^{hyperbolic\, secant\, temporal}\), which is normalized to \((1-R)E\), is:
3.4.8 Case 4: Lorentzian temporal distribution (normalized)
The probability density of the distribution with Gaussian x and y-spatial, uniform z-spatial and Lorentzian-temporal sub-distributions \({S}_{x,y,z\,spatial,normalized}^{lorentzial\, temporal}\left[\frac{W}{{m}^{3}}\right]\), which is normalized to \((1-R)E\), is:
\({S}_{x,y,z\,spatial,normalized}^{rectangular\,temporal}, {S}_{x,y,z\,spatial,normalized}^{gaussian\,temporal}, {S}_{x,y,z\,spatial,normalized}^{hyperbolic\,secant\,temporal} and {S}_{x,y,z\,spatial,normalized}^{lorentzial\,temporal}\) can be used in the Two-Temperature equations, in place of \(S\), as they show power per volume induced to the electrons due to the absorption of the laser irradiation.
4 State-of-the-Art Regarding the TTMs
Researchers have tried to solve all the above TTM types (PTS, HOS, HTS, DHTS, STS, DPL). In order to solve them, various computational methods (Analytical Methods, Finite Difference Method-FDM-, Finite Element Method-FEM-, Finite Volume Method-FVM-, Meshless Methods-MM-, Molecular Dynamics-MD- and Artificial Neural Networks-ANN-) have been used.
4.1 Parabolic One Step Model (POS)
Ho et al. [49] created a simulation to study the temperature distribution, evaporation rate and melting and ablation depth of Al, Cu and Au irradiated by a nanosecond laser. The model was solved numerically with Eulerian equations for mass, momentum, energy and isentropic gas equation and the results showed that surface temperature and the melting depth increase linearly proportionally as the laser fluence increases, whereas the ablation depth increases exponentially with laser fluence. Yilbas and Shuja [50] investigated the temperature distribution on three materials’ surfaces (Ni, Ta and steel) irradiated by a pulsed laser with pulse duration of 1.2, 1.3 and 1.4 μs, by creating a classical Fourier heat conduction model and solving it analytically. The estimated results were in a good agreement with the experimental data. Oane et al. [51] investigated analytically, with Zhukovsky formalism and Integral Transform Technique (ITT), the irradiation of Au by a nanosecond laser. The authors concluded that their method can be extended and to other laser processes and in laser additive manufacturing.
4.2 Parabolic Two Step Model (PTS)
In bibliography, many PTS models are solved with FDM. Koba and Dai [52] developed a stable FDM simulation to solve the PTS heat transfer equations regarding a micro-sphere irradiated by an ultrashort pulsed laser. However, because in FDM it is rather complicated to find a general exact solution, only some specific numerical examples have been analyzed in order to check the validity of the developed FDM. In these numerical examples, the electron and lattice temperature distributions on the sphere have been calculated for the following times: 0.2 ps, 0.25 ps, 0.5 ps, 1 ps and 2 ps. Zhai et al. [53] created a stable FDM, coupled with Knudsen number and Robin’s boundary condition, which is expressed by a coefficient, which shows the coincidence of the heat conduction model with the Boltzmann transport equation. This model was used to solve a PTS model of a metallic film irradiated by an ultrashort pulsed laser. To test the accuracy of the model, three numerical experiments were given, in which the electron and lattice temperatures were calculated for discrete times and for different values of Knudsen number and coefficient of coincidence. Furthermore, Shen et al. [54] developed a stable FDM to solve the PTS heat transfer model of a porous metallic film, irradiated by an ultrashort pulsed laser. Knudsen number, fractional-order time-derivative and the L1 approximation have been introduced to the PTS model, in order to lead to a tridiagonal linear solution system, which could be easily solved by the Thomas algorithm. Lastly, two numerical examples were given in order to test the accuracy of the FDM.
FEM is also widely used into solving the PTS model. Balasubramni et al. [55] studied with FEM the ablation of an Au film irradiated by a femtosecond laser. To study the ablation, firstly, electron and lattice temperatures were calculated via a PTS model, where thermophysical and optical properties were considered temperature-dependent. With this simulation, Au damage thresholds were estimated for different film thicknesses. The simulation results were then validated by experimental data. Fang et al. [56] created a PTS model with temperature dependent electron heat capacity, electron thermal conductivity, absorptivity and electron–phonon coupling factor. This temperature dependency was based on the electron Density of States (DOS) and the structure of the energy bands for Au and for temperatures over 104 K. The agreement of this simulation with experiments shows that DOS should be taken into consideration for temperatures over 104 K. Saghebfar et al. [40] developed a PTS model, which considered the ballistic range of the electrons, in order to study the effect of a femtosecond laser on a double-layer metallic film (Au/Cr). Because the electro-phonon coupling coefficient of Cr is higher than that of Au, the lattice temperature increased very fast at the interface between Au and Cr. On the other hand, Kapitza resistance between Au and Cr operated as a limiting factor that did not allow the overheating of the interface. The opposing mechanisms of electron–phonon coupling and Kapitza resistance led, eventually, to the interface equilibrium. Liu et al. [57] used a picosecond laser to print conductive Ag lines with the Laser Induced Forward Transfer (LIFT) method. To analyze the temperature distribution on the surface of the silver plate, a PTS model with a Gaussian source was created. The results of the simulation were in a good agreement with the experimental data. Schmidt et al. in [58] studied the ablation of Ag-, Al-, Fe- and Ni-surfaces irradiated by two sub-threshold pulses. The studies were numerical (PTS) and experimental. To estimate the ablation experimentally, a mass spectrometer was used. The results showed that there was a good agreement between simulation and experiments and that the electron thermal conductivity majorly affected the lattice temperature. Naldo et al. [59] proposed a combined experimental and numerical method, in order to estimate the electron–phonon coefficient G. Specifically, they used the experimental method of Time Domain Thermoreflectance (TDTR) in order to estimate the Reflectance vs Time curve of Au irradiated by a laser. Then, they created a PTS model and fine-tuned the G parameter till the Reflectance-Time curve of the simulation results best fitted to the experimental Reflection-Time curve, according to the method of least squares fitting.
Other computational methods have also been used to solve the PTS model. Yamashita et al. [60] coupled MD with PTS model in order to investigate the dominant heat transport mechanism, due to femtosecond laser irradiation, for various time intervals. To couple MD with PTS model, the kinetic energy of the atom at the specific position studied in MD, defined the lattice temperature of PTS model, according to:
The simulation showed that at the early stages of laser irradiation, electron heat conduction was the dominant heat transfer mechanism, whereas after some picoseconds, also the thermal shock wave mechanism started to appear. After electron-lattice equilibrium, both mechanisms were involved in an equal rate. Bora et al. [61] proposed a promising ANN to solve the PTS heat transfer equations for double-layer films irradiated by ultrashort pulsed lasers. The network was then tested for an Au/Cr double-layer film. Anisimov and Rethfeld [13] estimated the analytical solution of a PTS model for four limiting cases: \(a) t\ll {\tau }_{e},\) \(b) {\tau }_{e}\ll {\tau }_{p}\ll {\tau }_{i},\) \(c)for\,a\,laser\,beam\,distribution\,of\,Q(t)={Q}_{o}{\left(\frac{t}{{\tau }_{p}}\right)}^\frac{3}{2},\) where \({Q}_{o}\) is an initial theoretical laser heat flux and \(d) for\,{T}_{i}-{T}_{m}\ll {T}_{m}.\) Nicarel et al. [12] proposed a mathematical model to solve the PTS model for metallic films, such as Au, irradiated by an ultrashort laser. This model implemented the Zhukovsky formalism, which inserted unitless parameters with direct physical meaning, leading to a model that could easily be extended to other materials.
4.3 Hyperbolic One Step Model (HOS)
According to Cattaneo-Vernotte equation, Fourier law is less accurate in describing heat transfer due to ultrashort laser irradiation and thus electron relaxation time should be introduced in the equation. For this reason, Mochnacki and Paruch [25] developed a HOS heat transfer model to estimate the temperature profile of an Au film irradiated by an ultrashort laser. The simulation was solved with FDM. In this paper, τe was estimated experimentally. However, because these experiments were not easy, the authors, also, proposed an evolutionary algorithm that predicted τe.
4.4 Hyperbolic Two Step Model (HTS)
Qiu and Tien [15] used the Mac Cormack scheme of FVM with an uniform grid system in order to solve the HTS model. The results were in a good agreement with experimental data. Liu [62] developed a numerical method, which was based on Laplace transformation and Gaussian elimination algorithm, in order to solve the HTS heat transfer equations for multi-layer films irradiated by ultrashort lasers. To check the accuracy of the numerical model, the analytical equations for two-layer and three-layer films were, also, estimated. Tunc et al. [63] created an HTS model in order to study numerically the irradiation of two-layer films by an ultrashort laser. The two-layer films were Au/Si, Au/Ni, Au/W, Au/Al and Au/Pb. The results showed that the electron temperatures on the Au-surface were rather low, due to the high value of gold’s G. Moreover, it was found that as the laser intensity increased and the pulse duration decreased, the electron temperature increased. Furthermore, the thermal damage threshold was higher, when the substrate layer was a material of high G.
4.5 Dual Hyperbolic Two Step Model (DHTS)
Chen et al. [64] created a FDM simulation to solve the DHTS equations for Au films. The model was coupled with material removal equations, which described the thermal (melting and evaporation) and non-thermal (hot electron blast) mechanisms. Dai and Niu [65] developed a FDM to solve the DHTS heat transfer equations for a double-layer (Au/Cr) film irradiated by an ultrashort pulsed laser. The results showed that for very small interfacial contact between Au and Cr, the temperature jump was very high.
4.6 Semiclassical Two Step Model (STS)
Ren et al. [66] created a STS model to study the irradiation of Au films by an ultrashort laser. The equations of the STS model were solved with FDM. The results showed that the STS model predicted lower values of electron and lattice temperatures, which were more accurate than the results of parabolic and hyperbolic TTMs, especially for high laser fluence and pulse duration.
4.7 Dual Phase Lag Model (DPL)
Tzou [38] proposed the DPL model. This universal model could examine the physical responses of heat conduction both into the micro- and macro-scale. In order to prove the validity of the model, the author solved a one-dimensional example by developing a numerical algorithm, which was based on the Laplace inversion integral, the Fourier integral and the Riemann sum approximation. Yang [67] used the inverse DPL model to estimate the strength of an ultrashort pulsed laser with Gaussian temporal profile which irradiated a medium with constant thermal properties. The inverse DPL problem was solved with Finite Element Differential Method (FEDM), which is a combination of FEM and FDM. Dutta et al. [68] investigated the analytical solution of a DPL model, which described the irradiation of A6061 and Cu3Zn2 nanofilms by an ultrashort pulsed laser. The laser source was Gaussian and took into consideration the absorptivity and reflectivity of the films. The analytical model was based on the Duhamel’s theorem and the finite integral transform approach. The limitation of this study was that the numerical results for Al and Cu films were compared with the experimental results for gold, because experiments on Al and Cu could not be found.
4.8 Summary of the State-of-the-Art of All the TTMs
All the above papers are summed up into Table 1.
5 State-of-the-Art: Comparative Studies Between Two or More TTMs
Qiu and Tien [30] studied the irradiation of metallic films according to the classical heat conductivity (POS) and the PTS model. The models of the irradiation of an Au-surface by a nanosecond and a picosecond pulsed laser were solved numerically with FDM by the Crank–Nicholson method. The results showed that in the case of the picosecond laser POS model overpredicted the lattice temperature and underpredicted the heat-affected zone, whereas for the nanosecond laser no difference between the two models was observed. To validate the models, experiments were carried out. Specifically, the authors tried to estimate Te by measuring with a probe laser beam the reflectivity change of the metal’s surface. However, according to the authors the calculation of the optical properties involved high uncertainty, so the results may have low validity. Nevertheless, by comparing the picosecond laser simulations with the experiment, they observed that the PTS model gives a solution which is much closer to the experimental results. Qiu et al. [69] created a POS and a PTS model in order to investigate the thermical and mechanical responses of an Au-film irradiated by a nanosecond laser. The models were solved numerically with FDM by the Crank–Nicholson scheme and the results showed that for nanosecond pulse durations, POS was equally accurate as the PTS model. Hector Jr. et al. [70] created a PTS and an HTS model to examine the irradiation of a metallic film by a mode locked Nd:YAG laser, which had a very small pulse duration. The models were solved analytically and the results showed that for extremely short pulse durations, the PTS model severely underestimated the surface temperatures. On the other hand, the results of the HTS model were much more realistic. However, as the pulse frequency decreased, the differences between the PTS and HTS models also decreased. Majchrzak [26] developed a PTS and an HTS model to study the irradiation of a metallic film by a pulsed laser of 100 ps pulse duration. The results showed that for bigger values of laser pulse duration (in this case for 100 ps), there were some differences between the PTS and HTS models, regarding the electron and lattice temperatures, but these differences were not so severe as in the case of 0.1 ps for example. Liu et al. [71] solved with FDM two TTMs. The one model was a PTS and the other was an HTS. Then, they compared the simulation solutions with experimental data from literature. The results showed that both models gave abruptly the same Te. On the other hand, the damage threshold predicted by the HTS model was much closer to the experimental results, compared to the PTS model. Chen and Beraun [27] developed an DHTS model to investigate the irradiation of Au-films by an ultrashort laser. This model, also, included ballistic phenomena and temperature dependent thermophysical properties. Furthermore, the DHTS model was compared with the HTS and the PTS models and also with experimental data from other researchers. The results showed that the DHTS model was more accurate into predicting the thermal behavior and that the ballistic phenomena improved the prediction of the melting threshold fluence. Niu and Dai [72] solved with FDM a PTS and an DHTS model in order to study the ablation of a two-layer film irradiated by an ultrashort pulsed laser. It was observed that the PTS model estimated higher electron temperatures, lower ablation depths and higher normal stresses compared to the PTS model. Shen and Zhang [73] studied the irradiation of a 2D finite slab by a moving annular laser heat source. This phenomenon was studied with the POS, HOS and DPL models, which were solved analytically by utilizing the Green’s function approach and the method of superposition. The results showed that the DPL model gave much more accurate solutions to the problem compared to the POS and HOS models. Xu et al. [74] used FEDM coupled with the Chebyshev spectral element and the element mapping technique to solve and compare the HOS with the DPL heat conduction models of a plate irradiated by an ultrashort Gaussian laser. The results showed that the wave overlaps of the HOS model may lead to abnormalities (local temperatures below 0 K), whereas the DPL model led to much more realistic and accurate solutions.
5.1 Summary of the State-of-the-Art Regarding the Comparative Studies Between Two or More TTMs
All these studies that compare two or more TTM models are summarized below (Table 2).
6 Mind Map for the Selection of the Most Appropriate TTM
As observed from the literature above, many different TTMs exist and each one of them has its advantages and disadvantages. In order to help researchers to decide which one of these models is most appropriate regarding the parameters of each specific study, the following mind map is created (Fig. 2):
Mind Map for the Selection of the Most Appropriate TTM
The mind map above (Fig. 2) is a general guidance, which accrues from the comparison of the results of the existing bibliography. The categorization of pulse durations and fields of research and their connection with the different TTMs is based on what is considered as more acceptable from the majority of the existing bibliography. So, because the above mind map is only a general guidance, it does not mean that not following it, will necessarily lead to wrong results. However, this mind map can save valuable time from researchers, as they will be able to choose the most appropriate model according to what they study, so they will not need to do extended trials with different TTMs.
7 State-of-the-Art Regarding the Modeling of Ultrashort Pulsed Lasers and Recommendations
All the simulations above are modeling the laser sources with Gaussian temporal distributions. However, the experimental studies advocate those pulses generated from ultrashort lasers are not always Gaussian. Specifically, chirping, mode-locking and narrow linewidth techniques, used to produce ultrashort lasers can give different shapes, such as hyperbolic secant and Lorentzian [75]. Jain et al. [76] compared the effect of Gaussian, hyperbolic secant and Lorentzian temporal-profiles of a laser pulse on the transference of laser energy into the energy of the ions under strong radiation pressure regime. They found out that the Lorentzian pulse describes more accurately the energy transference to the ions. Terzic et al. [77] investigated the Compton scattering, which is observed at high laser intensities that are produced with a combination of chirping and higher order harmonics techniques. These incident laser pulses were modeled with Gaussian, Lorentzian and hyperbolic secant distributions.
Other researchers modelled laser sources with only hyperbolic secant temporal distributions. Halford [78] modelled the picosecond solid-state laser pulses with a hyperbolic secant equation, in order to study the effect of non-resonant dispersion in a propagation medium. Mollenauer et al. [79] modelled the single-mode picosecond Nd:YAG laser with hyperbolic secant distribution, in order to examine the narrowing of pulses close to the soliton period. In [80], Chen et al. used a hyperbolic secant equation to describe a sub-picosecond semiconductor monolithic passive Colliding Pulse Mode-locking (CPM) laser, which is widely involved in metal dying. Moreover, Piovella [81] showed that the irradiation emitted by a free electron laser was best approximated by a hyperbolic secant equation. Melinger et al. [82] studied the excitation of molecules at high vibration states by using an ultrashort pulsed laser, modelled with a hyperbolic secant distribution. Chan and Liu [83] investigated the effect of Raman scattering and third order dispersion on the optimum compression ratio and the compressed pulse width of a femtosecond mode-locked Nd:YAG laser. In order to model the laser, a hyperbolic secant equation was used. Kablukov et al. [84] developed a hyperbolic secant model to describe a Yb-doped fiber laser. The hyperbolic secant spectrum of the laser was then validated by spectral experiments with a cladding-pumped Yb-fiber laser. Shixing et al. [85] studied the transmission characteristics of a mode-locked laser. In this paper, the pulses were assumed to be hyperbolic secant. Singh et al. [75] applied optical rectification, which is widely used in Nd:YAG lasers, to pulses with hyperbolic secant shapes.
Some fewer researchers have modeled the laser sources with Lorentzian temporal distributions. Laser Induced Plasma (LIP) is widely used in microfabrication. LIP can be produced by Nd:YAG lasers [86]. According to Rosas-Roman et al. [87], lasers that produce LIP are best approximated by Lorentzian or combination of Gaussian and Lorentzian (also known as Voigt) equations. Petrovic and Miladinovic [88] compared the effect of Gaussian and Lorentzian laser beam profiles on the distribution of energy of ionized photoelectrons. The results were validated by experiments with a CO2 laser and the main conclusion of this study was that the spatial and temporal distribution of laser beams strongly affected the energy distribution.
As it is observed by the papers above, till now, hyperbolic secant and Lorentzian temporal laser beam models have, mainly, been used to study the lasers’ optical characteristics and properties. On the other hand, according to the authors’ best knowledge, no hyperbolic secant or Lorentzian temporal laser beam models were found to be used in laser-matter interaction simulations. Furthermore, no TTMs that use hyperbolic secant or Lorentzian temporal distributions for the laser sources were found. However, according to [44], hyperbolic secant and Lorentzian temporal distributions of laser beams may be more appropriate to accurately describe experimental situations. Thus, as future research, it would be recommended to use hyperbolic secant or Lorentzian temporal distributions for the modelling of the ultrashort pulsed laser sources of TTMs and compare the accuracy of the results with those of the Gaussian laser beam temporal profiles.
8 Theory Regarding the Implementation of Computational Methods on Heat Transfer Equations and the TTM
In all the above studies, in order to solve the Partial Deferential Equations (PDEs) that describe the TTM, various computational methods have been used. Some researchers solved the PDEs analytically, whereas others solved them numerically. The numerical methods used are: Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), Meshless Methods (MM), Molecular Dynamics (MD) and Artificial Neural Networks (ANNs).
All the above methods have their advantages and disadvantages. In order to choose which one method is the most appropriate for each different TTM problem, it is firstly necessary to understand how each method works. For this reason, the classical Fourier heat conduction Eq. (2) is solved below with all these different methods. For reasons of simplification, it will be assumed that \(\rho , {C}_{p}, k and S\) are constant.
8.1 Analytical Methods
As an example of how to solve a heat equation analytically, the below simplest possible form of this problem is solved:
with initial conditions: \(T\left(x,0\right)=f\left(x\right)\) and boundary conditions: \(T\left(0,t\right)=T\left(L,t\right)=0\).
As it is observed above, this is a very simple heat equation of a 1D rod without any heat sources (\(S=0)\). To solve this problem, the method of separation of variables [89] is used, so it is assumed that:
Thus, from the boundary conditions, it is calculated that:
Moreover,
So, by rewriting the heat equation with the separated variables:
To solve the \({X}^{{\prime}{\prime}}=const. \cdot X\), three cases are taken into consideration.
8.1.1 Case 1: \(const.={\lambda }^{2}>0 , \lambda >0\)
And from the boundary conditions:
8.1.2 Case 2: \(const.=0\)
and from the boundary conditions:
8.1.3 Case3: \(const.=-{\lambda }^{2}<0 , \lambda >0\)
and from the boundary conditions:
So,
Moreover,
Given that \(T\left(x,t\right)=X(x)\overline{T }(t)\) and assuming that \({A}_{n}={B}_{n}{C}_{n}\) then:
From the initial conditions:
It is known from mathematics that:
So,
By substituting (91) to (88), eventually the analytical solution is:
As it is observed, even for the simplest form of heat Eq. (1D, zero heat sources and boundary conditions), the calculations are rather complicated. Now, given that TTM problems involve the combination of various phenomena (convection, radiation, heat sources, ablation, conjunction of Te and Ti, etc.), the complexity increases drastically and only some highly simplified versions of these problems can be solved analytically. Furthermore, these analytical calculations require deep mathematical knowledge, which sometimes is out of the interests of an engineer. On the contrary, in the engineering and industrial world, lasers are used in very complex manufacturing processes (with complex geometries and combination of various phenomena), which makes the calculations of the TTMs with only analytical methods impossible. For this reason, various numerical methods have been used to solve the TTM problems.
8.2 Numerical Methods
8.2.1 Finite Difference Method (FDM)
The basic idea behind FDM is to discretize the control volume/surface of the problem with a grid of a finite number of points. As an example, the following 2D grid (Fig. 3) is given:
Grid of discretized points of an FDM model
In this example, the space of the control surface has been discretized by (I + 1) points (with a distance of Δx in between them) according the x-axis and (J + 1) points (with a distance of Δy) according the y-axis. Each of these points has its own temperature. For the random point (i,j), its temperature is symbolized as Ti,j.
For reasons of better visual understanding, the above example gives a 2D grid. However, the grid and the symbol of temperature can be generalized in order to include and the discretization according the z-axis. If it is assumed that the control volume is discretized with (K + 1) points according the z-axis, then the temperature at the random point (i,j,k) is symbolized as Ti,j,k.
Now, if the temperature of each point changes with time, then it is necessary to, also, add a time discretization to the problem. Usually, in the simulation the time discretization is implemented as an appropriate time step that has been set by the researcher. If it is assumed that the duration of the phenomenon is from 0 to L, then the temperature at the random point (i,j,k) at the random time l is symbolized as Ti,j,k,l.
In order to understand how the FDM solves a problem, the following 3D heat equation with a constant heat source is given as an example:
To solve this equation with FDM [90], the following Taylor expansions (explicit method) are needed:
The above Taylor expansion is based on the explicit method. If it is desired to use the implicit method then the only difference is that the “l” should be substituted by “l + 1” in the Taylor expansions of \(\frac{{\partial }^{2}T}{\partial {x}^{2}}, \frac{{\partial }^{2}T}{\partial {y}^{2}} and \frac{{\partial }^{2}T}{\partial {x}^{2}}\). Except from the explicit and implicit methods, also other methods have been developed, such as Crank–Nicholson [91], which lead to more accurate results, but are more complicated.
By discretizing the 3D heat equation with the explicit Taylor expansions, then:
Some of the temperatures of the above discretized equation are known (initial and boundary conditions), whereas some other temperatures are unknown, so they need to be predicted as a first step and then with iterations to be corrected. These iterations will continue till the correct temperature is found for each point and for each time step.
As it is observed in the process above, in FDM, the control volume is discretized by a set of nodes. The results (temperatures) are calculated only for these nodes. An advantage of this method is that it requires generally less computational power than that of other numerical methods. On the other hand, the non-continuity of the heat flux due to the points-discretization is a severe disadvantage, which makes the solution less realistic. Finally, although the FDM grid is pretty accurate at discretizing rectangular geometries and very-large-scale simulations, it cannot easily handle curved boundaries. Thus, it is difficult to accurately simulate ablation with FDM.
8.2.2 Finite Element Method (FEM)
In order to understand how FEM works, the same heat Eq. (93), as with FDM, will be solved now with FEM. This equation can be equally written as:
\(-\)
To apply the FEM, the above equation is implemented and weighted with a test function φ [91]:
According to the divergence theorem [92], applied to \(q\varphi\):
However,
So,
By substituting \(\underset{\Omega }{\int }\left(\nabla \cdot {\varvec{q}}\right)\varphi dV\) to (100):
The test function φ most commonly is a polynomial. However, it can also be another function or have a constant value.
In order to solve a problem with FEM, firstly the control surface (2D FEM) or the control volume (3D FEM) should be discretized by mesh elements. In order to calculate the temperature of each node, the above Eq. (104) should be solved for each mesh element h, as shown in Fig. 4.
Triangular FEM mesh
The Fig. 4 above shows an example of a 2D triangular mesh elements discretization. To calculate the temperature at an internal node (for example the red node), the following equation is used [93]:
In this case, φ is zero for all the elements, except from the surrounding “pink” elements. So, only these elements support the “red” node and only for these nodes the equation above is calculated.
On the other hand, to calculate the temperature at a boundary node (for example the green node), the following equation is used [94]:
In this case, φ is zero for all the elements, except from those with an edge (in 2D) or a face (in 3D) on the boundary. So, for example the green node is supported only by the green elements and not by the blue.
As it is observed above, FEM is a very symmetric method and the flux boundary conditions can directly and naturally be imposed, so there is no need for any computationally expensive reconstruction methods, in contrast with FVM (as it will be analyzed below). This is a major advantage of this method. On the other hand, the main disadvantage of the method is that the global statement for FEM (over the whole domain) guarantees only global conservation, but not local conservation. This means that the net flux is in balance only over the domain boundaries and not over the boundaries of each element. In some cases, this may lead to unrealistic results (for example temperatures below 0 K) [95].
8.2.3 Finite Volume Method (FVM)
In order to understand how FVM works, the same as previously heat Eq. (93) is solved. To find out the equations that will be used by the FVM, the same process is followed as with FEM (Eq. (100)-(104)). The only difference to these equations is that in FVM, the weak formulation is a special case of the Eq. (104) of FEM. Specifically, in FVM, φ = 1, so the FVM weak formulation is [96]:
To apply the FVM, firstly, the control surface or the control volume should be discretized into cells. In the example below, the cells are triangular (Fig. 5). Moreover, in contrast with FEM, in FVM each cell is assumed to be an individual domain Ωh. So, in FVM, the temperature calculated by the Eq. (107) is estimated in a “theoretical” point in the middle of the cell.
Triangular FVM cells
As it is observed, in FVM heat flux is estimated by the same equation for both internal and boundary cells. However, because the temperature is estimated only for the center point of each cell, the boundaries of each cell are not well defined. To solve this problem, FVM needs to be completed by a reconstruction method, which typically is a local interpolation between the neighboring cells. The disadvantage of the reconstruction method is that increases the computational complexity and time of the simulation, which is the main drawback of the FVM. On the other hand, the advantage of FVM is that it solves the heat equation for each cell (local statement), which leads to local conservation, as the net flux in each cell is always in balance.
8.2.4 Meshless Methods (MMs)
In order to understand how MMs work, it is important, firstly, to introduce what the Radial-Basis Function Interpolation (RBF) [97] is. Temperature T (as any other field variable) can be expanded by a finite number of functions:
where \(a\) is the expansion coefficient, which is defined by least squares method or direct collocation of the temperature at the data centers and \(X\) is an expansion function.
Commonly, in MMs, RBF are used as expansion functions. Many different types of RBF exist (polyharmonics splines, Gaussian, multiquadrics), however in this MM example, the inverse Muliquadrics RBF is used:
where c is a curve shape parameter, which is defined ad-hoc and \({r}_{j}\) is the Euclidean distance of a field point from the data center and is estimated for a given t as:
where \({x}_{j},\) \({y}_{j}, {z}_{j}\) are the data center coordinates.
In order to understand how MMs work, the same heat Eq. (93), as previously, is solved. Firstly, a finite number of NB and NI data centers [98] are assumed on the boundary and inside the domain, respectively (Fig. 6):
Schematic representation of MMs data centers
According to (108), temperature T can be written for NB + NI data centers as:
So, by substituting the above function (111) to the heat Eq. (93):
Then, the same substitutions should be applied and to the boundary conditions. Afterwards, these relations should be solved with iterations regarding the chosen time step, in order to calculate the temperature of each data center for all the time intervals.
In MMs, a type of connection (mesh, grid) between the nodes is not required, thus all the extensive properties (for example mass and kinetic energy) are applied into single nodes and not into mesh elements. This may lead to ill-conditioned equations, so it is necessary to choose appropriate interpolation functions, such as RBF. However, these expansion functions increase drastically the computational time, which is the main disadvantage of the method. On the other hand, the absence of mesh makes it easier to solve Lagrangian equations, where the nodes can move around regarding the velocity field. This is a huge advantage for the solution of fluid dynamics and large-deformations problems, such as the movement of the melting pool and the simulation of the ablation, respectively.
8.2.5 Molecular Dynamics (MD)
For the microscopic solution of the heat transfer Eq. (93), usually MD are combined with a macroscopical method (FEM, FDM, FVM) described previously. As an example, in this review paper, MD are combined with FEM [99] (Fig. 7).
Schematic representation of the combination of MD with FEM
In Fig. 7, MD particles (controlled only be the MD system) are shown as black dots, whereas the open circles (controlled only by the FEM system) represent the nodes of the elements (each blue triangle is one element) and the black nodes into open circles represent the combined node-particle (controlled by both FEM and MD systems).
In FEM, the researcher sets a time step ΔtFEM and the solver estimates the T through the heat Eq. (93) for each node of each element and for each ΔtFEM. Then, for the node-particles, the temperature calculated for each node and for each ΔtFEM, is transformed into MD-particle velocity through the equation [98]:
where \({k}_{B}\) is the Boltzmann constant, \({m}_{p}\) is the MD-particle mass and \({v}_{p}\) is the MD-particle velocity.
For the MD simulation, researcher should set another time step ΔtMD, which is usually smaller than ΔtFEM, so that it can estimate the MD-particle motion. Then, in order to estimate the movement of the neighbor MD-particles (black dots without open circles), a potential function that takes as reference point the node-particle is used. Usually, in TTM the potential function is an electric potential that combines the position of the desired MD-particle with the position of the neighbor particle-node. Then, the movement for both MD-particles and node-particles is estimated for each ΔtMD regarding the Newton motion equation, which is integrated according to the Verlet integration method (which is explained later). When all the ΔtMD that are included in one ΔtFEM are estimated, then the new temperature for each node is calculated with FEM and the same process as previously is repeated for the calculation of the particles’ movement for the new ΔtFEM interval (Fig. 8).
Algorithm followed for the calculations of the combined MD/FEM model
To better understand how MD work, the basic theory of MD computations is given below. Specifically, to deeper understand the mechanism of heat transfer, knowledge of the microscopical phenomena is also necessary. However, extending the macroscopic analysis to microscopic spatial and temporal conditions is an inefficient method. Thus, in order to investigate the microscopic phenomena, it is necessary to introduce the Molecular Dynamics (MD) simulations.
In MD, the Newton’s equation of motion is solved for each molecule/atom/particle:
where \({\varvec{r}}, {\varvec{F}}\) are the position vector and the force vector, respectively and the index \(p\) refers to the p-th particle.\(\Phi\) is the potential energy of the system, which can be estimated by:
where \(\varphi\) is the effective pair potential and \({r}_{pq}\) is the distance between the p-th and q-th particles.
The simplest way to estimate the pair potential is with the Lennard–Jones potential function:
where \(\sigma\) and \(\varepsilon\) are length and energy scales, respectively.
This function is appropriate to express the potentials of monoatomic molecules (Ne, Ar, Kr, Xe). However, this function is not so accurate into expressing the potentials of metallic molecules. Although, estimation of system potential through pair-potentials is the simplest method, its validity is questionable. Thus, it is necessary to move to many-body potentials for the estimation of the heat transfer system potential. The most appropriate system potential for metals, which is calculated via many-body potentials is the Embedded Atom Method (EAM) [100]:
where \(H\) is the embedded function and it represents the energy needed to move particle p into the electron cloud, \({\rho }^{a}\) is the spherically averaged atomic electron density and \(\varphi\) is the electrostatic pair-potential, which can be estimated regarding the Coulomb potentials:
where \({\varepsilon }_{o}\) is the vacuum permitivity and \(Z\) is the electron charge.
Thus, from (117) and (118) \(\Phi\) can be written as:
So, from (114) and (119) Newton equation of motion is written:
Then (120) is solved numerically with Verlet integration. The Verlet integration uses the following algorithm [101]:
The implementation scheme of this algorithm uses the following steps:
-
(1)
(1) Calculation of
$$\left. {\frac{{d{\varvec{r}}_{{\varvec{p}}} }}{dt}} \right|_{{t + \frac{1}{2}\Delta t_{MD} }} = \left. {\frac{{d{\varvec{r}}_{{\varvec{p}}} }}{dt}} \right|_{t} + \frac{1}{2}\left. {\frac{{d^{2} {\varvec{r}}_{{\varvec{p}}} }}{{dt^{2} }}} \right|_{t} \Delta t_{MD}$$(123) -
(2)
(2) Calculation of
$${\varvec{r}}_{{\varvec{p}}} \left( {t + \Delta t_{MD} } \right) = {\varvec{r}}_{{\varvec{p}}} \left( t \right) + \left. {\frac{{d{\varvec{r}}_{{\varvec{p}}} }}{dt}} \right|_{{t + \frac{1}{2}\Delta t_{MD} }} \Delta t_{MD}$$(124) -
(3)
(3) Derivation of
$$\frac{{(d^{2} r_{p} )}}{{(dt^{2} )}}|_{{(t + \Delta t_{MD} )}} from\,\phi \,using\,r_{p} (t + \Delta t_{MD} )$$ -
(4)
Calculation
$$\left. {\frac{{d{\varvec{r}}_{{\varvec{p}}} }}{dt}} \right|_{{t + \Delta t_{MD} }} = \left. {\frac{{d{\varvec{r}}_{{\varvec{p}}} }}{dt}} \right|_{{t + \frac{1}{2}\Delta t_{MD} }} + \frac{1}{2}\left. {\frac{{d^{2} {\varvec{r}}_{{\varvec{p}}} }}{{dt^{2} }}} \right|_{{t + \Delta t_{MD} }} \Delta t_{MD}$$(125)
The major advantage of MD simulations is that they increase the microscopic understanding of heat transfer phenomena, such as the three-phase interface (solid surface, liquid melting pool and evaporation), nucleation theory, shock wave propagation, etc. that cannot be studied with any macroscopic methods, discussed previously. The main limitation of MD is that it requires a great amount of time, in order to give results, as very short time steps are taken (in order to be able to describe phenomena with duration of some fs) and a very large number of MD-particles are needed [102].
8.2.6 Artificial Neural Network (ANN)
Many different ANNs have been developed. To solve the heat transfer partial differential equation, the most well-known ANN method is the Physics Informed Neural Nets (PINNs). In this example, PINNs will be used to solve the heat Eq. (93). ANNs learn better when the equations are dimensionless, so the following variables are introduced [103]:
where \({T}_{o}, \left({x}_{L}, {y}_{L}, {z}_{L}\right)\,and \,{t}_{L}\) are reference temperature, lengths and time, respectively.
By replacing these variables to (93) the following dimensionless equation is acquired:
However, for computational simplicity in this example, from now on \({x}^{*}\) will be the generalized length that will symbolize all the lengths regarding the x, y and z-axis. Moreover, in this example, \({T}^{*}\) is the only unknown, thus only one ANN is needed, called NN. Also, it is assumed that the inputs of NN are \({x}^{*}\) and \({t}^{*}\) and the output is \({u}^{*}\), which represents the temperature. A simple NN, which consists of only 3 hidden layers and 2 hidden units per hidden layer is given in Fig. 9.
Schematic representation of an ANN
In this NN diagram (Fig. 9), \({z}_{i}^{j}\) is the output of the \(i-th\) hidden unit of the \(j-th\) hidden layer, \({W}_{k,i}^{j}\) symbolizes the weight with which the output \({z}_{k}^{j-1}\) is multiplied and the result of this multiplication is, then, added to the sum of the \(i-th\) hidden unit of the \(j-th\) hidden layer, \({b}_{i}^{j}\) is the bias, which is added to the \(i-th\) hidden unit of the \(j-th\) hidden layer, \(\Sigma\) is the sum symbol and \(\sigma\) is the activation function, which is applied to the sum of the node’s inputs. Activation function can be linear, sigmoid, hyperbolic tangent, etc. However, usually in heat transfer problems the hyperbolic tangent function is used:
where \(y\) is the sum of the node’s inputs.
This NN can be generalized for L hidden layers and M hidden units in each hidden layer. Moreover, in order to symbolize the inputs and the outputs, it is assumed that the x* input is the 1st unit of the 0th layer, t* input is the 2nd unit of the 0th layer and u* is the 1st unit of the (L + 1) layer. Thus, u* can be expressed as:
where
where:
During the training of the ANN, each pair of x* and t* inputs give a u* output. This u*(x*, t*) output is compared to the value of T*(x*, t*). This comparison is carried out through a loss function. This process is given schematically in Fig. 10. In this example, loss function \({J}_{LOSS}\) is calculated as:
where N is the number of training points selected and \({t}_{des}^{*}\) is the desired time for which the loss function is calculated.
Schematic representation of NN training process
\({J}_{LOSS}\) is estimated for each \({t}^{*}\) of the training set. Then, an optimization method, such as Adam optimization [104] or L-BFGS [105], is used to optimize the weights and biases. The analysis of these optimization methods is out of the targets of the current review paper and thus the reader is referred to the corresponding citations. The ANN is executed again and again with the newest weights and biases in each repetition till the loss function goes below the set error threshold. Moreover, via trial-and-error tests by the researcher, the optimum number of hidden layers and hidden units should be determined.
The main advantages of the usage of ANN to solve the heat transfer equation or the TTM problems are the ability of ANN to work with incomplete knowledge and produce data to complete the missing information and their ability for machine learning, as ANN are trained to make decisions on their own for similar events. On the other hand, the limitation of ANN is the requirement for many training points, which may not be available or they may need very expensive experiments in order to be acquired. Furthermore, the accuracy of ANN in TTM problems is questionable, as Te cannot be acquired experimentally and thus the training set is based only on simulation results acquired with FDM, FVM, FEM, etc. However, all numerical methods contain errors on their own, so the combination of the numerical errors with the ANN-errors leads to highly inaccurate results. Thus, it is suggested to use ANN only for some very general predictions and then to move on to a numerical method for more details [106].
8.3 Mind Map for the Selection of the Most Appropriate Computational Method for Solving TTMs
As observed, many different computational methods to solve TTMs exist and each one of them has its advantages and disadvantages. So, to help researchers decide which one of these methods is more appropriate regarding the targets of each specific study, a mind map (Fig. 11) is created.
Mind map for the selection of the most appropriate computational method for solving TTMs
However, it should be noted that this mind map (Fig. 11) is only a general guidance, which accrues from the comparison of the results of the existing bibliography. The categorization of the computational targets and their connection with the different computational methods that solve TTMs is based on what is considered as more acceptable from the majority of the existing bibliography. So, because this mind map is a general guidance, it does not mean that not following it will, necessarily, lead to wrong results. However, this mind map can save valuable time from researchers, as they will be able to choose the most appropriate computational method according to their computational targets, so they will not need to do extended trials with different computational methods.
9 Conclusions
The current work is a review paper, which presents, analyzes and compares all the basic TTMs, the ultrashort pulsed lasers models and the computational method, used to solve the TTM equations, appeared into the literature.
-
From the comparison of the different TTMs it is concludes that PTS is more appropriate for \(10ps\le {t}_{p}\le 1ns\), whereas all the PTS, HTS, DHTS, STS and DPL can be used for \({t}_{p}\le 10ps\). However, for \({t}_{p}\le 10ps\) the accuracy of the results increases as moving from PTS to HTS and to DHTS. Furthermore, models, such as STS and DPL can also study the electron drift velocity and macroscopical parameters, respectively.
-
From the analysis of the different ultrashort pulsed laser sources it is shown that the laser spot can be described spatially with a Gaussian distribution regarding the x and y-axis, an uniform distribution that follows the Beer-Lambert law regarding the z-axis and temporally with either rectangular, Gaussian, hyperbolic secant or Lorentzian distributions.
-
From the comparison of all the different computational methods, it is shown that analytical methods can be used for oversimplified TTMs, ANNs are appropriate only for rough temperature predictions, FDM is suitable for fast computations, FEM and FVM give highly accurate and realistic results, MM are used when large deformations and/or fluid dynamics are included and MD are appropriate for microscopical studies, such as the movement of the particles.
The novelty of the current review paper is that it organizes and visualizes all these conclusions regarding the different TTMs and laser source models into mind maps that can be easily used by researchers to help them choose the most appropriate TTM and laser source model for their specific case study parameters. Furthermore, another novelty of the current review paper is that it makes recommendations for future work regarding the more accurate and realistic modelling of the ultrashort pulsed laser sources.
References
Dubey AK, Yadava V (2008) Laser beam machining-A review. Int J Mach Tools Manuf 48(6):609–628. https://doi.org/10.1016/j.ijmachtools.2007.10.017
Wise S (2013) Laser Fundamentals. GIS Fundamental. CRC Press, Boca Raton
Eichhorn M (1989) Laser Physics, From Principles to Practical Work in the Lab. Springer, Switzerland
Svelto O (1999) Principles of lasers. Sens Rev 19(4):325–326
Nolte S, Schrempel F, Dausinger F (2016) Ultrashort Pulse Laser Technology. Springer, Amsterdam
Kiran Kumar K, Samuel GL, Shunmugam MS (2019) Theoretical and experimental investigations of ultra-short pulse laser interaction on Ti6Al4V alloy. J Mater Process Technol 263:266–275. https://doi.org/10.1016/j.jmatprotec.2018.08.028
Sobolev SL (2016) Nonlocal two-temperature model: application to heat transport in metals irradiated by ultrashort laser pulses. Int J Heat Mass Transf 94:138–144. https://doi.org/10.1016/j.ijheatmasstransfer.2015.11.075
Caceres CA (1976) New technology demands teamwork. Hospitals (Lond) 50(7):103–107
Ayars E (2006) An introduction to computer simulation methods: applications to physical systems (third edition). Am J Phys 74(7):652–653. https://doi.org/10.1119/1.2219401
Holman V (1999) Heat Transfer (10th edition). McGraw-Hill, New York
Gamaly EG, Rode AV (2013) Physics of ultra-short laser interaction with matter: from phonon excitation to ultimate transformations. Prog Quantum Electron 37(5):215–323. https://doi.org/10.1016/j.pquantelec.2013.05.001
Nicarel A, Oane M, Mihailescu IN, Ristoscu C (2021) Fourier two-temperature model to describe ultrafast laser pulses interaction with metals: A novel mathematical technique. Phys Lett A 392:127155. https://doi.org/10.1016/j.physleta.2021.127155
Anisimov S, Rethfeld B (1997) On the theory of ultrashort laser pulse interaction with a metal. Nonres Laser-Matter Interact 3093(1):192–203
Hohlfeld J, Wellershoff SS, Güdde J, Conrad U, Jähnke V, Matthias E (2000) Electron and lattice dynamics following optical excitation of metals. Chem Phys 251(1–3):237–258. https://doi.org/10.1016/S0301-0104(99)00330-4
Qiu TQ, Tien CL (1993) Heat transfer mechanisms during short-pulse laser heating of metals. J Heat Transfer 115(4):835–841. https://doi.org/10.1115/1.2911377
L’Annunziata MF (2016) Cherenkov. Radioactivity. Elsevier, Amsterdam
Singh N (2004) Relaxation between electrons and surface phonons of a homogeneously photoexcited metal film. Pramana - J Phys 63(5):1083–1087. https://doi.org/10.1007/BF02704347
Kaganov M, Lifshitz I, Tanatarov L (1957) Relaxation between Electrons and the Crystalline Lattice, J. Exptl. Theoret. Phys., 4(2):232–237
Allen PB (1987) Theory of thermal relaxation of electrons in metals. Phys Rev Lett 59(13):1460–1463. https://doi.org/10.1103/PhysRevLett.59.1460
Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61(1):41–73. https://doi.org/10.1103/RevModPhys.61.41
Brorson SD, Fujimoto JG, Ippen EP (1987) Dynamics in thin gold films. Phys Rev Lett 59(17):1962–1965
Maurer MJ (1969) Relaxation model for heat conduction in metals. J Appl Phys 40(13):5123–5130. https://doi.org/10.1063/1.1657362
Cattaneo C (1958) A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus 247:431–433
Vernotte P (1958) Les paradoxes de la theorie continue de l’equation de la chaleur. Compte Rendus 246:3154–3155
Mochnacki B, Paruch M (2013) Cattaneo-Vernotte equation identification of relaxation time using evolutionary algorithms. J Appl Math Comput Mech 12(4):97–102. https://doi.org/10.17512/jamcm.2013.4.12
Majchrzak E (2012) Parabolic and hyperbolic two-temperature models of microscopic heat transfer: Comparison of numerical solutions. Mater Sci Forum 706–709:1454–1459
Chen JK, Beraun JE (2001) Numerical study of ultrashort laser pulse interactions with metal films. Numer Heat Transf Part A Appl 40(1):1–20. https://doi.org/10.1080/104077801300348842
Chen JK, Tzou DY, Beraun JE (2006) A semiclassical two-temperature model for ultrafast laser heating. Int J Heat Mass Transf 49(1–2):307–316. https://doi.org/10.1016/j.ijheatmasstransfer.2005.06.022
Ashcroft N, Mermin I (1976) Solid State Physics. Harcourt, Orlando
Qiu TQ, Tien CL (1992) Short-pulse laser heating on metals. Int J Heat Mass Transf 35(3):719–726. https://doi.org/10.1016/0017-9310(92)90131-B
Jiang L, Tsai HL (2005) Improved two-temperature model and its application in ultrashort laser heating of metal films. J Heat Transfer 127(10):1167–1173. https://doi.org/10.1115/1.2035113
Liboff R (2003) Kinetic Theory Classical, Quantum, and Relativistic Descriptions. Springer, New York
Jiang L, Tsai HL (2004) Prediction of crater shape in femtosecond laser ablation of dielectrics. J Phys D Appl Phys 37(10):1492–1496. https://doi.org/10.1088/0022-3727/37/10/012
Jiang L, Tsai HL (2005) Energy transport and material removal in wide bandgap materials by a femtosecond laser pulse. Int J Heat Mass Transf 48(3–4):487–499. https://doi.org/10.1016/j.ijheatmasstransfer.2004.09.016
Yu Tzou D (1993) An engineering assessment to the relaxation time in thermal wave propagation. Int J Heat Mass Transf 36(7):1845–1851. https://doi.org/10.1016/S0017-9310(05)80171-1
Bai C, Lavine AS (1995) On hyperbolic heat conduction and the second law of thermodynamics. J Heat Transfer 117(2):256–263. https://doi.org/10.1115/1.2822514
Zhang Z (2020) Nano and Microscale Heat Transfer. Springer, Switzerland
Tzou DY (1995) A unified field approach for heat conduction from macro- to micro-scales. J Heat Transfer 117(1):8–16. https://doi.org/10.1115/1.2822329
Wang L, Zhou X, Wei X (2007) Heat conduction: mathematical models and analytical solutions. Springer, Berlin
Saghebfar M, Tehrani MK, Darbani SMR, Majd AE (2017) Femtosecond pulse laser irradiation of gold/chromium double-layer metal film: The role of interface boundary resistance in two-temperature model simulations. Thin Solid Films 636:464–473. https://doi.org/10.1016/j.tsf.2017.06.032
Masina BN, Bodkin R, Mwakikunga B, Forbes A (2011) Laser beam shaping for studying thermally induced damage. Laser Beam Shap XII 8130:81300H. https://doi.org/10.1117/12.898514
Chien SK, Tsai KT, Li YH, Wu YT, Chen WL (2020) A simple and accurate 3d numerical model for laser cladding. J Glob Hist 36:39–60. https://doi.org/10.1017/jmech.2019.30
Caiazzo F, Alfieri V (2018) Simulation of laser heating of aluminum and model validation via two-color pyrometer and shape assessment. Materials (Basel) 11(9):1506. https://doi.org/10.3390/ma11091506
Shchatsinin I (2009) Free Clusters and Free Molecules in Strong Shaped Laser Fields. Dissertation, Freie Universität Berlin
Advanced kinetics, “Mathematical description of ultrashort laser pulses,” pp. 1–8, 2016, [Online]. https://www.ethz.ch/content/dam/ethz/special-interest/chab/physical-chemistry/ultrafast-spectroscopy-dam/documents/lectures/advancedkineticsFS16/Exdef/advkin_exercise10.pdf.
Zinn-Justin J (2004) Gaussian Integrals. In: Zinn-Justin J (ed) Path Integr Quantum Mech. Oxford University Press
Boreman J (2013) On behalf of the fish. Fisheries 38(8):343. https://doi.org/10.1080/03632415.2013.813484
Sharif A, Farid N, Wang M, Choy KL, O’Connor GM (2022) The role of fluence in determining the response of thin molybdenum films to ultrashort laser irradiation; from laser-induced crystallization to ablation via photomechanical ablation and nanostructure formation. Appl Surf Sci 592:153315. https://doi.org/10.1016/j.apsusc.2022.153315
Ho JR, Grigoropoulos CP, Humphrey JAC (1995) Computational study of heat transfer and gas dynamics in the pulsed laser evaporation of metals. J Appl Phys 78(7):4696–4709. https://doi.org/10.1063/1.359817
Yilbas BS, Shuja SZ (1997) Heat transfer analysis of laser heated surfaces—conduction limited case. Appl Surf Sci 108(1):167–175. https://doi.org/10.1016/S0169-4332(96)00570-3
Oane M, Mihailescu IN, Sava B (2019) The linearized fourier thermal model applied to Au nanoparticles 1D and 2D lattices under intense nanoseconds laser irradiation pulses. J Mater Sci Eng 08(01):1–6. https://doi.org/10.4172/2169-0022.1000513
Kaba IK, Dai W (2005) A stable three-level finite difference scheme for solving the parabolic two-step model in a 3D micro-sphere heated by ultrashort-pulsed lasers. J Comput Appl Math 181(1):125–147. https://doi.org/10.1016/j.cam.2004.11.024
Zhai S, Huang L, Weng Z, Dai W (2020) Parabolic two-step model and accurate numerical scheme for nanoscale heat conduction induced by ultrashort-pulsed laser heating. J Comput Appl Math 369:112591. https://doi.org/10.1016/j.cam.2019.112591
Shen S, Dai W, Cheng J (2020) Fractional parabolic two-step model and its accurate numerical scheme for nanoscale heat conduction/. J Comput Appl Math 375:112812. https://doi.org/10.1016/j.cam.2020.112812
Balasubramni T, Kim SH, Jeong SH (2009) Enhanced two temperature modeling of ultrashort laser ablation for the investigation of thermionic emission characteristics. Appl Surf Sci 255(24):9601–9604. https://doi.org/10.1016/j.apsusc.2009.04.080
Fang R, Wei H, Li Z, Zhang D (2012) Improved two-temperature model including electron density of states effects for Au during femtosecond laser pulses. Solid State Commun 152(2):108–111. https://doi.org/10.1016/j.ssc.2011.10.031
Liu Q, Xu B, Zhang Y, Wang X, Mei X, Wang X (2021) Picosecond laser sintering of silver paste printed by laser induced forward transfer. Opt Laser Technol 135:106712. https://doi.org/10.1016/j.optlastec.2020.106712
Schmidt V, Husinsky W, Betz G (2002) Ultrashort laser ablation of metals: pump-probe experiments, the role of ballistic electrons and the two-temperature model. Appl Surf Sci 197(198):145–155. https://doi.org/10.1016/S0169-4332(02)00318-5
Naldo SB, Bernotas AV, Donovan BF (2020) Understanding the sensitivity of the two-temperature model for electron-phonon coupling measurements. J Appl Phys 128(8):085102. https://doi.org/10.1063/5.0019719
Yamashita Y, Yokomine T, Ebara S, Shimizu A (2006) Heat transport analysis for femtosecond laser ablation with molecular dynamics-two temperature model method. Fusion Eng Des 81(8):1695–1700. https://doi.org/10.1016/j.fusengdes.2005.09.011
Bora A, Dai W, Wilson JP, Boyt JC (2021) Neural network method for solving parabolic two-temperature microscale heat conduction in double-layered thin films exposed to ultrashort-pulsed lasers. Int J Heat Mass Transf 178:121616. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121616
Liu KC (2007) Analysis of thermal behavior in multi-layer metal thin films based on hyperbolic two-step model. Int J Heat Mass Transf 50(7–8):1397–1407. https://doi.org/10.1016/j.ijheatmasstransfer.2006.09.018
Tunç KMM, Günay ME, Bayata F (2020) “Analysis of the thermalization dynamics of two-layer thin films irradiated by femtosecond laser. Optik (Stuttg) 208(164137):2020. https://doi.org/10.1016/j.ijleo.2019.164137
Chen JK, Beraun JE, Grimes LE, Tzou DY (2002) Modeling of femtosecond laser-induced non-equilibrium deformation in metal films. Int J Solids Struct 39(12):3199–3216. https://doi.org/10.1016/S0020-7683(02)00242-1
Dai W, Niu T (2008) A finite difference scheme for solving a nonlinear hyperbolic two-step model in a double-layered thin film exposed to ultrashort-pulsed lasers with nonlinear interfacial conditions. Nonlinear Anal Hybrid Syst 2(1):121–143. https://doi.org/10.1016/j.nahs.2007.07.001
Ren Y, Chen JK, Zhang Y (2012) Modeling of ultrafast phase changes in metal films induced by an ultrashort laser pulse using a semi-classical two-temperature model. Int J Heat Mass Transf 55(5–6):1620–1627. https://doi.org/10.1016/j.ijheatmasstransfer.2011.11.018
Yang CY (2014) Estimation of the heat source of laser pulses by a dual-phase-lag model. Int J Heat Mass Transf 73:358–364. https://doi.org/10.1016/j.ijheatmasstransfer.2014.01.054
Dutta J, Biswas R, Kundu B (2020) Analytical solution of dual-phase-lag based heat transfer model in ultrashort pulse laser heating of A6061 and Cu3Zn2 nano film. Opt Laser Technol 128:106207. https://doi.org/10.1016/j.optlastec.2020.106207
Qiu T, Tien C, Shannon M, Russo R (1994) Gold Films During Nanosecond Laser Pulse. J Experimental Heat Transfer, 7:175–188
Hector LG, Kim WS, Ozisik MN (1992) Hyperbolic heat conduction due to a mode locked laser pulse train. Int J Eng Sci 30(12):1731–1744. https://doi.org/10.1016/0020-7225(92)90091-T
Liu Y, Li L, Lou Q (2019) A hyperbolic lattice Boltzmann method for simulating non-Fourier heat conduction. Int J Heat Mass Transf 131:772–780. https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.110
Niu T, Dai W (2009) A hyperbolic two-step model based finite difference scheme for studying thermal deformation in a double-layered thin film exposed to ultrashort-pulsed lasers. Int J Therm Sci 48(1):34–49. https://doi.org/10.1016/j.ijthermalsci.2008.02.001
Shen J, Zhang X (2022) Analysis of dual-phase-lag heat conduction in a two-dimensional slab heated by a moving annular laser pulse. Appl Math Model 105:160–178. https://doi.org/10.1016/j.apm.2021.12.024
Xu BB, Gao XW, Cui M (2021) High precision simulation and analysis of non-Fourier heat transfer during laser processing”. Int J Heat Mass Transf 178:121574. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121574
Singh RK, Singh M, Rajouria SK, Sharma RP (2017) High power terahertz radiation generation by optical rectification of a shaped pulse laser in axially magnetized plasma. Phys Plasmas 24(10):103103. https://doi.org/10.1063/1.4994296
Jain V, Soni KK, Maheshwari KP, Malav H, Jaiman NK (2015) Effect of laser-pulse-shape on the transference of laser energy into the energy of fast ions in radiation pressure dominant regime. Indian J Pure Appl Phys 53(12):785–789
Terzić B, Reeves C, Krafft GA (2016) Combining harmonic generation and laser chirping to achieve high spectral density in Compton sources. Phys Rev Accel Beams 19(4):1–8. https://doi.org/10.1103/PhysRevAccelBeams.19.044403
Halford C (1971) The effect of non-resonant dispersion on short hyperbolic secant laser pulses. Physics Letters, 35A(5):357–358
Mollenauer LF, Tomlinson WJ, Stolen RH, Gordon JP (1983) Extreme Picosecond Pulse Narrowing Via the Soliton Effect in Single-Mode Optical Fibers. Optics Letters, 8(5): 204. https://doi.org/10.1364/cleo.1983.thq4.
Chen YK, Wu MC, Tanbun-Ek T, Logan RA, Chin MA (1991) Subpicosecond monolithic colliding-pulse mode-locked multiple quantum well lasers. Appl Phys Lett 58(12):1253–1255. https://doi.org/10.1063/1.104327
Piovella N (1991) A hyperbolic secant solution for the superradiance in free electron lasers. Opt Commun 83(1–2):92–96. https://doi.org/10.1016/0030-4018(91)90528-L
Melinger JS (1995) Selective excitation of vibrational overtones in an anharmonic ladder with frequency- and amplitude- modulated laser pulses. Choice Rev Online 33(9):33–1251
Chan KC, Liu HF (1995) Short pulse generation by higher order soliton-effect compression: effects of optical fiber characteristics. IEEE J Quantum Electron 31(12):2226–2235. https://doi.org/10.1109/3.477751
Kablukov SI, Zlobina EA, Podivilov EV, Babin SA (2012) Output spectrum of Yb-doped fiber lasers. Opt Lett 37(13):2508. https://doi.org/10.1364/ol.37.002508
Shixing T, Liling Y, Xincai Z, Teng H, Jianzhang L, Qixian P (2013) Nonlinear effects of hyperbolic secant pulse transmission in fiber. Infrared and Laser Engineering, 42(10): 9–11
Zakuskin AS, Labutin TA (2022) Spectrochimica Acta Part B : atomic spectroscopy processing of thomson scattering spectra for diagnostics of laser-induced plasma. Spectrochim Acta Part B At Spectrosc 190:106394. https://doi.org/10.1016/j.sab.2022.106394
Maldonado JL (2017) Spectrochimica Acta part B simultaneous height adjust fitting : An alternative automated fitting procedure for laser-induced plasma spectra composed by multiple Lorentzian profiles. Spectrochim Acta Part B At Spectrosc 134:1–5. https://doi.org/10.1016/j.sab.2017.05.004
Petrović VM, Miladinović TB (2014) Influence of the spatial and temporal distribution of an incident laser beam profile on the energy distribution of ionized photoelectrons. J Exp Theor Phys 119(4):651–656. https://doi.org/10.1134/S1063776114100082
Wanxie Z, Xiangxiang Z (1993) Method of separation of variables and Hamiltonian system. Numer Methods Partial Differ Equ 9(1):63–75. https://doi.org/10.1002/num.1690090107
Wicaksana A (2016) Conservative finite-difference methods on general grids. CRC Press, Boca Raton
Jin B, Li B, Zhou Z (2018) An analysis of the crank-nicolson method for subdiffusion. IMA J Numer Anal 38(1):518–541. https://doi.org/10.1093/imanum/drx019
Markvorsen S (2006) The direct Flow parametric Proof of Gauss ’ Divergence Theorem revisited. pp. 1–24, [Online]. https://orbit.dtu.dk/en/publications/the-direct-flow-parametric-proof-of-gauss-divergence-theorem-revisited(197c67b5-6a5e-4d9d-866e-1ad50052fafb).html.
Zienkiewicz O, Taylor R, Zhu J (2005) “Finite Element Method for Solid and Structural Mechanics. Elseiver, Amsterdam
Bunting CF (2008) Introduction to the finite element method”. IEEE Int Symp Electromagn Compat. https://doi.org/10.1109/ISEMC.2008.4652216
Pentenrieder B (2005) Finite Element Solutions of Heat Conduction Problems in Complicated 3D Geometries Using the Multigrid Method. Diploma Thesis, Technische Universitat Munchen
Eymard R, Gallouet T, Herbin R (2000) Finite Volume Methods. In: Ciarlet P, Lions J, Handbook of Numerical Analysis, 7th edn, Elsevier, Amsterdam, pp. 713–1018
Wright G (2003) Radial Basis Function Interpolation: Numerical and Analytical Developments. Dissertation, University of Colorado
Patel VG, Rachchh NV (2019) Meshless method—Review on recent developments. Mater Today Proc 26:1598–1603. https://doi.org/10.1016/j.matpr.2020.02.328
Smirnova JA, Zhigilei LV, Garrison BJ (1999) Combined molecular dynamics and finite element method technique applied to laser induced pressure wave propagation. Comput Phys Commun 118(1):11–16. https://doi.org/10.1016/S0010-4655(98)00175-1
Daw MS, Foiles SM, Baskes MI (1993) The embedded-atom method : a review of theory and applications. Mater Sci Rep. 9:251–310
Vorob’ev IN (2011) Molecular dynamics method for proteins with ionization-conformation coupling and equilibrium titration. Mol Biol (Mosk) 45(2):346–355
Li Q, Liu C (2012) Molecular dynamics simulation of heat transfer with effects of fluid-lattice interactions. Int J Heat Mass Transf 55(25–26):8088–8092. https://doi.org/10.1016/j.ijheatmasstransfer.2012.08.045
Jain AK, Mao J, Mohiuddin KM (1996) Artificial neural networks: a tutorial. Computer 29(3):31–44. https://doi.org/10.1109/2.485891
Kingma DP, Ba JL, (2015) “Adam: A method for stochastic optimization,” 3rd Int. Conf. Learn. Represent. ICLR 2015—Conf. Track Proc., pp. 1–15
Liu T, Li D (2007) Convergence of the BFGS-SQP method for degenerate problems. Numer Funct Anal Optim 28(7–8):927–944. https://doi.org/10.1080/01630560701405002
Mijwil MM (2018) Artificial Neural Networks Advantages and Disadvantages. LinkedIn, no. March, pp. 1–2, [Online]. https://www.linkedin.com/pulse/artificial-neural-networks-advantages-disadvantages-maad-m-mijwel/.
Funding
Open access funding provided by HEAL-Link Greece.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Alexopoulou, V.E., Markopoulos, A.P. A Critical Assessment Regarding Two-Temperature Models: An Investigation of the Different Forms of Two-Temperature Models, the Various Ultrashort Pulsed Laser Models and Computational Methods. Arch Computat Methods Eng 31, 93–123 (2024). https://doi.org/10.1007/s11831-023-09974-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-023-09974-1