Thermoelastic Vibration of Nickel Film Irradiated by Femtosecond Laser: Molecular Dynamics Study

Abstract

A detailed understanding of the physical essence of the interaction between a femtosecond laser and its target material remains an important and challenging goal. In this paper, the thermoelastic vibration behavior of nickel films irradiated by a femtosecond laser is studied by a molecular dynamics method combined with a two-temperature model. The model fully defines the spatial distribution of laser energy, the photoelectron coupling, and the electron-lattice coupling, and elucidates the temperature and stress evolution within the nickel film under femtosecond laser irradiation. Furthermore, the whole process and the mechanism of thermoelastic vibration is revealed at the atomic level. The thermoelastic vibration is divided into two stages, including continuous expansion during the process of energy relaxation and periodic expansion and contraction after reaching thermal equilibrium. The elastic oscillation of thin films is driven by periodic changes in energy, including the energy of atomic thermal motion and collective atomic motion. The effect of pulse fluence on thermoelastic vibration is also discussed in detail to provide reasonable suggestions for limiting this effect. This study provides the theoretical foundation and a feasible method for a deeper understanding of the interaction mechanisms between femtosecond lasers and materials.

1. Introduction

Laser-induced damage of materials by femtosecond laser pulses has been a subject of study for many years as ways to change surface morphology [1,2,3,4,5]. In terms of femtosecond laser–metal interactions, the target usually exhibits two ablation morphologies: thermal ablation and nonthermal ablation [6]. This is related to the Gaussian distribution of laser energy in space. Nonthermal ablation occurs in the small fluence range of the laser spot edge and is usually regarded as an unwanted thermal effect. Although material is not removed, its structure and morphology are changed. The accumulation of thermal effects can lead to the formation of defects in the material. Therefore, it is necessary to limit nonthermal ablation in femtosecond laser-related applications. Nonthermal ablation is a series of structural changes and phase transformation in materials induced by femtosecond lasers at low fluence. During this process, the surface of the material usually shows thermal expansion and a small amount of melting. These behaviors are attributed to the laser-induced stress wave generated by rapid heating [7,8].

The theory of the coupling between the heating of the material produced by the laser and the stress produced by thermal dilatation has been extensively studied, both from analytical and numerical viewpoints [9,10]. Wang et al. formulated a generalized solution to the thermoelastic wave in a semi-infinite solid induced by pulsed laser heating [11]. The solution considered the non-Fourier effect in heat conduction, the coupling effect between temperature and strain rate, and the volumetric absorption of laser beam energy. Detailed information on laser-induced (from nanosecond to femtosecond) thermoelastic waves on nickel films was provided. In 1997, Maznev et al. utilized the one-dimensional equation of thermoelasticity to describe the lattice deformation of gold films under 150 ps laser irradiation, with the time-resolved thermoelastic vibrations being clearly demonstrated [12]. Similarly, Gusev also developed a continuous model to describe the laser-induced thermoelastic effect [13]. All these studies provided feasible methods for understanding the laser-induced ultrafast thermoelastic dynamics of metal films. However, the nonequilibrium effect of the electronic lattice caused by the femtosecond laser was not considered in the above study. In subsequent studies, a two-step model accounted for the nonequilibrium state between electrons and the lattice applied to metals illuminated with ultrafast pulsed lasers [14]. The generation, propagation, and attenuation of thermoelastic waves induced by pico- and femtosecond laser pulses were clarified. However, this previous analysis was based on a continuum model, and the dynamic behavior of the materials was not considered. Analysis is much more difficult when both melting and ablation occur because there are many nonlinear phenomena involved [15,16]. On the one hand, the thermal and elastic properties change when the material melts [17]. On the other hand, melting and ablation are accompanied by energy conversion inside the material. Due to the existence of latent heat during melting, the atomic mechanical energy decreases, which may lead to decreased thermal expansion. Therefore, the dynamic behavior of the materials must be carefully considered when describing femtosecond laser-induced stress waves.

The definition of dynamic behaviors requires atomic-level information. In this regard, molecular dynamics simulations have, from a mathematical viewpoint, the advantage of including nonequilibrium behavior [18,19]. According to the details of the interaction potential used, nonlinearities such as the latent heats of melting and vaporization and the dependence of thermal and elastic constants on temperature are automatically taken into account.

Based on molecular dynamics, this work provides a detailed characterization of the thermoelastic vibration behavior in the nonthermal ablation region of nickel thin films under femtosecond laser-induced stress. The atomic image is established, and the melting, structural transformation and energy transformation of materials are confirmed. At the same time, several laser fluences are applied to examine the effect of pulse fluence on thermoelastic vibration. Furthermore, qualitative guidance for controlling the unnecessary thermal effect is proposed based on the obtained results.

2. Computational Models

In the present study, a nickel film was selected to calculate the thermoelastic vibration under the action of a femtosecond laser. Nickel-based coatings are widely used to improve the corrosion resistance of various components. Therefore, during the precision machining of components with nickel films, the interaction between the femtosecond laser and nickel film has always been a topic of interest [20,21,22]. The analysis of nickel film thermoelastic vibration can help control the surface structure and remove unnecessary thermal effects.

A molecular dynamics method coupled with a two-temperature model (TTM-MD) is employed in this research. The TTM is used to describe the deposition of laser energy, the electron-phonon energy coupling and the fast electron heat conduction, while the subsequent thermodynamic behaviors of the material are described by the MD model. Compared with the traditional continuous level model, the TTM-MD model provides a more accurate and detailed description of laser-induced stress and material dynamic behaviors.

The TTM describes the heat conduction of electrons and lattices in a simulated system with two coupled equations of generalized heat conduction. In this paper, we only use the equation of electronic heat conduction, which can be expressed as follows [23]:

$${\mathrm{C}}_{\mathrm{e}}(\partial {\mathrm{T}}_{\mathrm{e}}/\partial \mathrm{t})={\mathrm{k}}_{\mathrm{e}}({\partial }^2{\mathrm{T}}_{\mathrm{e}}/\partial {\mathrm{z}}^2)-\mathrm{g}({\mathrm{T}}_{\mathrm{e}}-{\mathrm{T}}_{\mathrm{l}})+\mathrm{S}(\mathrm{z},\mathrm{t})$$

(1)

$$\mathrm{S}(\mathrm{z},\mathrm{t})={\mathrm{I}}_0((1-\mathrm{R})/{\mathrm{L}}_{\mathrm{p}})\exp [-4\ln (2)((\mathrm{t}-{\mathrm{t}}_0)/{\mathsf{\tau }}_{\mathrm{L}}{)}^2]\exp (-\mathrm{z}/{\mathrm{L}}_{\mathrm{p}})$$

(2)

where T_e and T_l represent the electron and lattice temperatures, respectively. C_e is the electron heat capacity, whose value depends on the electron temperature, and its specific expression is C_e = γT_e. k_e is the electrical thermal conductivity, the value of which is related to both the electron temperature and the lattice temperature, and its specific expression is k_e = k₀T_e/T_l. g is the electron-phonon coupling coefficient. S represents the laser fluence with a Gaussian temporal profile, and the specific form is shown in Equation (2), where I₀ is the peak power intensity, R is the material reflectivity, L_p is the optical absorption length, and τ_L is the full width at half maximum (FWHM) pulse duration. The values of the parameters for nickel are given in

Table 1. The values of the parameters used in the TTM-MD model.

Parameter	Value	References
γ constant	1065 J/m3K2	[7,21,27]
k0 constant	91 W/mK	[7,21,27]
g electron-phonon coupling	3.6 × 1017 W/m3K	[7,21,27]
coefficient
R reflectivity	0.62	[7]
Lp optical absorption length	13.5 nm	[7]
τL FWHM pulse width	500 fs
D dissociation energy	0.4205 ev	[27]
rε equilibrium distance	0.278 nm	[27]
b constant	14.199 nm−1	[27]

. It should be noted that Zhigilei et al. corrected the dependence of electronic heat capacity and electron phonon coupling constant on electron temperature, resulting in a more accurate description of heat conduction and energy coupling [24]. Additionally, as reported by Tsibidis and Kudryashov, the variations of material optical properties, such as reflectivity and absorption coefficient, with pulse width and laser fluence have also been incorporated into the model [25,26]. Kudryashov et al. revealed in detail the variations of multiple optical and thermal properties of Al films with laser fluence, and provided the lattice temperature and crater depth, taking into account various factors [27]. The reflectivity and other characteristics exhibited significant changes with laser fluence, particularly in the high fluence range (above F_melt). The reported results provide a wealth of reference and guidance for the computational study of the interaction between femtosecond lasers and metal films. It is necessary to consider the above corrections for accurately quantifying the thermodynamic behavior of materials under femtosecond laser irradiation. In this study, the laser fluences discussed are relatively low, even less than F_melt, and thus, the variations in the thermal physical parameters are not significant. Additionally, the objective is solely to qualitatively demonstrate the thermoelastic vibration phenomenon of Ni films and to provide fundamental physical information regarding temperature and stress. Therefore, in order to simplify the calculation process, the correction of multiple physical quantities was not considered.

The lattice temperature is defined by the molecular dynamics method, which considers the energy coupling by adding a velocity equalizing force to the atomic force:

$${\mathrm{m}}_{\mathrm{i}}({\mathrm{d}}^2\overrightarrow{{\mathrm{r}}_{\mathrm{i}}}/\mathrm{d}{\mathrm{t}}^2)=\overrightarrow{{\mathrm{F}}_{\mathrm{i}}}+\mathsf{\xi }{\mathrm{m}}_{\mathrm{i}}\overrightarrow{{\mathrm{v}}_{\mathrm{i}}}$$

(3)

where

$$\mathsf{\xi }=\sum _{\mathrm{k}=1}^{\mathrm{n}}\mathrm{g}{\mathrm{V}}_{\mathrm{N}}({\mathrm{T}}_{\mathrm{e}}^{\mathrm{k}}-{\mathrm{T}}_{\mathrm{l}}^{\mathrm{k}})/(\mathrm{n}\sum _{\mathrm{i}}{\mathrm{m}}_{\mathrm{i}}(\overrightarrow{{\mathrm{v}}_{\mathrm{i}}}{)}^2)$$

(4)

where m and r represent the mass and position vectors of atoms, respectively, F_i represents the force acting on atom i, and v is the thermal velocity of atoms. In the summation process, k represents the kth atom, and n is the total number of atoms in the current calculation system. The equilibrium factor ξ depends on the sum of the kinetic energy of each layer of atoms, and it regulates the motion of atoms by means of external forces. Another key problem of the model used in this paper is the selection of a potential function, which describes the interaction between atoms and further determines the behavior of the materials. As a relatively simple potential function, the reliability of the Morse potential in calculating the interaction between metal atoms with a face-centered cubic structure (FCC) has been fully verified [28,29]. Therefore, the Morse potential is used to calculate the atomic interaction in nickel thin films, and its expression is [28]:

$$\mathsf{\Phi }(\mathrm{r})=\mathrm{D}[\exp (-2\mathrm{b}(\mathrm{r}-{\mathrm{r}}_{\mathsf{\epsilon }}))-2\exp (-\mathrm{b}(\mathrm{r}-{\mathrm{r}}_{\mathsf{\epsilon }}))]$$

(5)

where D is the total dissociation energy and r_ε is the equilibrium distance. The constant b in this equation determines the shape of the potential curve. Equation (5) describes the interaction potential between atoms, representing the changes in atomic distance under the action of force, and further energy changes. Therefore, the interatomic force, i.e., F_i in Equation (3), can be solved according to Equation (5), and the position change of the atoms (r_i in Equation (3)) can be further obtained. The values of the parameters for nickel in the Morse potential are also given in Table 1.

The leapfrog algorithm is used to solve the above TTM-MD model, and the spatial discretization step can be estimated based on the von Neumann stability criterion [26]. In the present study, the time step and space step of the molecular dynamics calculation are 1 fs and 0.5 nm, respectively. The simulation system is constructed as an FCC nickel atom structure, with a space size of 2.3 nm × 2.3 nm × 35.3 nm containing 19,600 atoms. The width and length scale are slightly larger than the two times critical distance (1.06 nm for Ni) of atomic self-action to ensure the correctness of atomic behavior. The depth is chosen as 35.3 nm, in which all possible phenomena and mechanisms can be clearly characterized. All the details of the simulation system are shown in Figure 1. Free boundary conditions are applied to the left and right crystal faces to simulate the thin film surface, while periodic boundary conditions are applied to the other crystal faces. Laser pulses occur from left to right into the simulation system. The purpose of applying periodic boundary conditions is to shorten the computation time, which is a widely used boundary condition. The calculation is carried out at the nanoscale, which is much smaller than the actual size of the femtosecond laser spot (usually tens of microns). Therefore, the Gaussian distribution of pulse energy on the incident plane is neglected and replaced by a uniform distribution. In addition, the pulse energy follows the Beer-Lambert law along the depth direction of the system and presents a Gaussian distribution on the time scale. Before laser irradiation, a 20 ps simulation was conducted within the established atomic system, with temperature (300 K), energy, and atomic configurations stabilizing from 2 ps onwards. Therefore, the computational system relaxes efficiently and reaches a state of equilibrium. All calculations are self-programmed in MATLAB, including the construction of atomic systems, the definition of physical quantities, and the solution of model control equations.

3. Results and Discussion

3.1. Temperature Evolution and Laser-Induced Stress

The evolution of electron and lattice temperature on the surface (laser incident surface) and bottom of the nickel film when the pulse width is 500 fs and the laser fluence is 2000 J/m² is displayed in Figure 2. The selection of fluence is based on the threshold information of two ablation mechanisms of nickel under femtosecond laser irradiation. In the previous calculation, the behavior of nickel films in a large energy range (for 1000 J/m² to 3500 J/m²) was characterized. For the thin film system (35.5 nm) studied in this paper, the nonthermal ablation threshold is determined as 1000 J/m², and the thermal ablation threshold is approximately 2500 J/m². Thermoelastic vibration occurs in the nonthermal ablation region. Therefore, the fluence range of 1000 J/m² to 2500 J/m² is effective for characterizing thermoelastic vibration. In this paper, 2000 J/m² is used to analyze the evolution of laser-induced stress waves and thermoelastic vibrations. This clearly illustrates the deposition of laser energy in the electrons in the conduction band, the rapid electron heat conduction, and the energy coupling between electron phonons. The absorption of laser energy results in a dramatic increase in the electronic temperature. Meanwhile, the Beer–Lambert law determines the spatial energy distribution, which decreases exponentially along the axis direction. Therefore, a strong temperature gradient is generated in the electron system within the film. By the end of laser irradiation (approximately 1.5 ps), the maximum temperature difference between the surface and bottom of the film is 2000 K. Subsequently, such a temperature gap is continuously reduced by rapid electron heat conduction. Afterwards, the electron temperature of the entire film is substantially the same at approximately 10 ps. The energy redistribution of the electronic system is accompanied by the electron–phonon energy coupling process. This leads to the increasing temperature of the lattice, as shown in Figure 2. Due to the rapid oscillation of the lattice temperature, it is difficult to directly compare it with the electronic temperature. Therefore, the average temperature within 20 fs (10 fs before and after) is used to represent the lattice temperature at a given time. At approximately 13 ps, the surface electron temperature and lattice temperature are nearly identical, with values of 3080 K and 3020 K, respectively. Accordingly, the system starts to approach thermal equilibrium from 13 ps onward.

The crystal lattice of the film is heated to 2000 K or even higher in 5 ps, as indicated in Figure 2. During such a short period of time, the film does not have enough time to expand, so the upper and lower surfaces of the rapidly heated film generate strong compressive stress and propagate to the interior in the form of waves. In this paper, the stress state within the nickel films is calculated according to Equation (6):

$$\mathrm{P}=\mathsf{\rho }\mathrm{R}{\mathrm{T}}_{\mathrm{l}}+{\displaystyle \frac{1}{6\mathrm{V}}}⟨{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\sum _{\mathrm{j}\ne \mathrm{i}}\overrightarrow{{\mathrm{F}}_{\mathrm{ij}}}\cdot \overrightarrow{{\mathrm{r}}_{\mathrm{ij}}}⟩$$

(6)

where ρ is the atomic density, V is the original cell volume, and F and r are the force vector and position vector, respectively. This setup is derived from virial theorem, which considers the interaction of molecules in matter to derive the equation of state. The first part (ρRT) is from the momentum transport when atoms enter the domain where pressure is to be evaluated. This part is similar to the pressure for ideal gases, where the forces between molecules are neglected. The second part is from the interaction between molecules and can be considered as the average of the normal stresses on all the planes in three dimensions. In this work where the focus is on the thermodynamic processes during the laser ablation, Equation (6) is applicable as it considers the kinetic part of the pressure.

As shown in Figure 3, a strong compressive stress (yellow region) is formed in the nickel film from 0 ps to 7 ps. When the compressive stress wave reaches the free surface, it is reflected as a tensile stress wave and continues to propagate inward. According to the basic theory of stress waves, the stress wave will reflect the opposite stress state on the free surface, and the magnitude is consistent. Therefore, the film exhibits a tensile stress state (blue region) from 7 ps to 15 ps. A second reflection is generated when the wave reaches the free surface, and this cycle reciprocates. The alternating change of the yellow region (compressive stress) and the blue region (tensile stress) in Figure 3 illustrates this phenomenon. Whether it is compressive stress or tensile stress, its peak appears at the center of the nickel film and is approximately 1.5 GPa.

3.2. Thermoelastic Vibration and Atomic Structure Transformation

The discussion in Section 3.1 shows that the nickel film is always under complex thermomechanical interactions. To further characterize the behavior of the nickel film, atomic level information is given. The motion of atoms in the nickel film is exhibited in the snapshots of the simulation (Figure 4). After 10 ps, a small fraction of the heterogeneous melt layer can be observed in the vicinity of the upper and lower surfaces of the film because the laser energy deposition causes the temperature of the whole system to increase until it exceeds the melting point (approximately 2500 K for Morse-Ni). However, the solid–liquid interface moves slowly, and the melting depth is only approximately 5 nm within 40 ps, which indicates that the contribution of heterogeneous melting to the overall melting process of metals is very limited, which is consistent with previous studies [19]. In fact, the temperature stress coupling of the film is not enough to destroy the stability of the lattice. In this respect, homogeneous nucleation of the liquid phase cannot take place within the film. Therefore, the front of the solid–liquid interface can only move slowly from the surface to the interior of the film via heterogeneous nucleation. The velocity of the melting front propagation has a strong dependence on the degree of superheating [7]. As shown in Section 3.1, the film is balanced at approximately 3000 K, and such a small degree of superheat is insufficient to make the melting front move rapidly.

A more interesting result is that the nickel film undergoes expansion and contraction behavior, also known as thermoelastic vibration. This thermoelastic vibration behavior can be illustrated in two stages. As shown in Figure 4, in the first stage, the film expands continuously along the depth direction from 1 ps to 13 ps, and the total expansion is approximately 4 nm. In this stage, the film is still in energy relaxation, and the temperature has not yet reached equilibrium. At 13 ps, the nickel film changes from expansion to contraction, which lasts approximately 7 ps and ends at 20 ps. The total amount of expansion is reduced from 4 nm to 2.5 nm, which indicates that the shrinkage process is very limited and that the entire film is still in an expanded state relative to the initial state. It is noteworthy that, in the subsequent time, the nickel film continues to change between expansion and contraction, so the second stage of thermoelastic vibration is a process of periodic expansion and contraction.

On the other hand, the atomic structure inside the film also shows periodic changes. In this paper, the atomic structure inside the nickel film is characterized by the structural order parameter (SOP), which can be obtained by the Fourier transform of the local density function. The SOP is a physical quantity used to measure the degree of structural disorder, with a value of 1 for ideal crystals and 0 for ideal fluids. The specific form is as follows:

$$\mathrm{S}(\mathrm{K})={\displaystyle \frac{1}{\mathrm{N}}}\left|{\sum }_{\mathrm{j}=1}^{\mathrm{N}}\exp (\mathrm{i}\overrightarrow{\mathrm{K}}\cdot \overrightarrow{{\mathrm{r}}_{\mathrm{j}}})\right|$$

(7)

where K is the reciprocal vector, and r is the position vector. It should be noted that the atomic system was first layered before the calculation. Due to the close relationship between S and atomic position vectors, calculating the S of each atomic layer can obtain its distribution along the depth direction. Therefore, N represents the number of atoms in each layer, rather than the number of atoms in the entire system.

The SOP values of the nickel film at various times are shown in Figure 5. The surface and bottom show a fairly low level (between 0–0.5) due to heterogeneous melting, which characterizes the disorder of the atomic structure. The disordered structure expands towards the inside of the film (up to 5 nm at 40 ps) as the solid–liquid interface migrates, which is consistent with the heterogeneous melting zone. Different from the surface, the SOP values of atoms within the nickel film show fluctuating changes. As shown in Figure 5a, at 13 ps, the SOP value within the film is at a low level, especially in the range of 10 nm to 20 nm (approximately 0–0.4). However, at 20 ps (Figure 5b), the SOP value increases to approximately 0.8, which is close to the ordered atomic structure. Remarkably, the changes are repeated from 30 ps to 40 ps, suggesting that the atomic structure within the film changes periodically.

3.3. The Mechanism of Thermoelastic Vibration

In fact, the film is always in the state of energy conservation, so its expansion and contraction necessarily correspond to the transformation of energy in the system. The energy deposition on the system occurs in three forms: the energy of atomic thermal motion, the energy that goes to the latent heat of melting and the energy of the collective atomic motion. The energy spent on melting is determined by the fraction of the liquid phase in the system and the latent heat of melting of the model material, which is not discussed in detail in this paper. Specific details of melting energy can be found in reference [30]. Compared with the energy of melting, we focus on the other two energies related to atomic motion. Figure 6a shows the changes in the energy of the thermal motion of atoms and the energy of the collective atomic motion in the nickel film as a function of time, which is obtained based on the virial theorem. In this paper, the above two kinds of energy are normalized. The energy of the two parts does not increase after the end of the energy deposition process but fluctuates in the time after the thermal balance, indicating the mutual transformation between the two parts. In comparing Figure 6a with Figure 4 and Figure 5, it can be seen that the time point of energy conversion is always consistent with the thermoelastic vibration and structural change of the film. When the energy of atomic thermal motion is converted to the energy of atomic collective motion, the film expands, and its structure appears disorderly. In contrast, when the energy is reversed, the film shrinks, and its structure is restored.

In addition, the temperature of the nickel film is based on the statistics of the atomic thermal motion velocity. Therefore, the variation of the energy of atomic thermal motion must cause the change of temperature, while the temperature information of the film surface does not display this trait (Figure 1). In contrast, we observed fluctuations in the temperature evolution of atoms within the film (Figure 6b), and the observed positions were consistent with those of the atomic structure. In fact, it is the conversion of energy that drives the change of atomic structure, so they both occur in the same region within the film. This indicates that energy conversion occurs within the film rather than on the surface.

In fact, the energy conversion between atoms in nickel film is affected by the stress state. The existence of tensile stress will drive the energy of thermal atomic motion to transform into the energy of concentrated atomic motion, thus driving the expansion of the nickel film, while the compressive stress acts in the opposite manner. This conclusion can be verified by comparing Figure 6a with Figure 3. Moreover, the greater stress within the film than on the surface also provides direct evidence for previous discussions on the location of energy conversion.

However, from 1 ps to 7 ps, the film under compression stress still expands continuously, which seems to contradict the previous analysis. In fact, the thermoelastic vibration of the film is the result of the coupled thermomechanical effect. The deposition of pulse energy in the initial time leads to a continuous increase in the concentrated motion energy of the atoms, and the film produces intense expansion. The expansion from 7 ps to 13 ps is the result of energy deposition and tensile stress. After 13 ps, the energy relaxation of the film almost ends, and the entire system reaches thermal equilibrium. It can be seen in Figure 7 that the velocity of atoms in the film follows a Maxwell–Boltzmann distribution at 13 ps. At 10 ps, the atomic velocities are more concentrated within the lower speed range. Particularly, the normalized velocity corresponding to the peak of the distribution probability is approximately 0.28. In contrast, at 13 ps and 20 ps, the distribution probability of higher normalized velocities is greater, particularly within the range of 0.4 to 0.8. Such a result suggests that at 10 ps, the atomic system is still in the process of heating up. It is noteworthy that there are also slight differences in the velocity distributions at 13 ps and 20 ps, indicating that even after thermal equilibrium is reached, the atomic velocities remain in a state of dynamic change. Although the peak distribution probabilities for both are observed at 0.32, the probability at 20 ps is slightly greater than that at 13 ps. This indicates that the temperature of the system at 20 ps is slightly higher than at 13 ps, which is consistent with the changes in temperature and the periodic thermoelastic vibrations. After thermal equilibrium, the effect of laser-induced stress relaxation dominates, leading to periodic expansion and contraction of the film. Therefore, the thermoelastic vibration of nickel film under femtosecond laser irradiation is divided into two stages, which are bounded by thermal equilibrium.

In terms of thermoelastic vibration, similar behavior has also been reported in related calculations. Wang et al. studied the development of displacement and stress during laser heating as well as their propagation and reflection in a 173 nm argon system [31]. The displacement at the surface is positive (towards outside) when a tensile stress emerges in the near surface region (7–15 ps). Subsequently, when the stress changes to the compression state (15–20 ps), a negative displacement occurs. Due to the large gap in materials, a quantitative comparison is difficult. However, the results of this paper are qualitatively consistent with previous studies.

On the other hand, the structural changes related to the laser-induced stress wave in the nickel film were observed in related experiments. Murphy and Schrider observed thin film removal from glass substrates after the irradiation of nickel films with femtosecond laser pulses [20,32]. For 20 nm thick films, material was removed within the film (intrafilm separation) at 2000 J/m². This result was attributed to launching a shock wave and accompanying rarefaction wave into the liquid nickel. The local pressure within the liquid becomes negative as the rarefaction wave forms and vacancies homogeneously coalesce into voids along a plane within the melt. This theory is consistent with the explanation of atomic structure transition in the present study. However, although the fluence used in this paper is the same as that in Reference [20], the homogeneous nucleation of holes and the eruption of the liquid layer did not take place within the film. Only structural changes were identified. This may be due to different film thicknesses. The necessary conditions for hole nucleation in nickel films have been determined in relevant calculations. The temperature and stress required for the homogeneous nucleation of voids and the eruption of the liquid layer were approximately 3000 K and 6 GPa, respectively [7]. Therefore, in the calculation of this paper, the coupling effect of temperature and tensile stress is insufficient to induce the nucleation of the voids. For the 35.3 nm nickel film, the same phenomenon as in the experiment may be observed by applying a higher fluence.

3.4. Effect of Fluence on Thermoelastic Vibration

Energy conversion is the key factor in the thermoelastic vibration of nickel film, and the energy of the collective atomic motion is directly related to the fluence. Therefore, the effects of the fluence of the femtosecond laser pulse on the thermoelastic behavior of the film are also discussed. In this work, 1250 J/m², 1500 J/m², 1750 J/m², 2000 J/m² and 2250 J/m² were selected for calculation, and the evolution of expansion rates with time corresponding to different fluences was obtained (Figure 8). The energy absorption and the formation of tensile stress make the expansion rate of the film increase continuously in the initial several picoseconds. Then, the energy relaxation ends, and the change in stress state promotes the occurrence of thermoelastic vibration, which shows that the expansion rate fluctuates with time. The film exhibits the same qualitative behavior at different laser fluences. However, the expansion rate at high fluence is always greater than that at low fluence, which has been shown from the initial expansion process. The variation of the initial expansion rate at different fluences is shown in Figure 9a. When the fluence increases by 1000 J/m², the initial expansion rate increases by approximately 80%. This is because the film has a higher energy of atomic concentration movement at high fluence during the initial energy deposition process.

The larger expansion rate means that the stress wave in the film may require a longer propagation time to reach the free surface to reflect and then affect the time of thermoelastic vibration. In this paper, the time between the peaks of the film expansion rate is defined as the period of thermoelastic vibration with a time scale of approximately dozens of picoseconds. As seen in Figure 9a, when the fluence increases from 1250 J/m² to 2250 J/m², the thermoelastic vibration period increases by approximately 6 ps, which indicates that the frequency of thermoelastic motion of the film decreases. The reasons for this phenomenon may be attributed to two factors: (i) The strengthening of fluence increases the expansion rate of the film, making the propagation time of the stress wave in the film longer. (ii) With the increase in fluence, the melting of the film surface will be intensified, and the liquid region will increase, while the propagation velocity of the stress wave in the liquid phase is lower than that in the solid phase. Figure 9b shows the fraction of the liquid phase in the nickel film at different fluences. The fraction of the liquid phase increases with increasing pulse fluence, and this trend is more obvious at high fluence. At low fluence, from 1250 J/m² to 1500 J/m², the volume fraction of the liquid phase increased only approximately 1.5%. However, from 2000 J/m² to 2250 J/m², the volume fraction of the liquid phase increased by approximately 8%. This change trend of the liquid volume fraction can be explained from two aspects. On the one hand, the increase in fluence means that more energy is deposited into the electrons and lattice, resulting in a material with a higher temperature. More large superheating is formed on the surface of the material, and the melting front moves towards the interior of the material faster. Therefore, the heterogeneous melting region on the film surface increases with increasing fluence. On the other hand, the increase in the liquid volume fraction may be related to the transformation of the melting mechanism. High temperature and high tensile stress lead to lattice instability. A large number of lattices collapse in a short time, resulting in the occurrence of homogeneous melting.

The thermoelastic vibration exists in a small energy range (nonablation region) at the edge of the spot, which results in the formation of a nanosurface structure at the edge of the processed region. Compared with the nonirradiated area, the machined edge shows poor surface roughness. As shown in Figure 9a, the thermoelastic vibration amplitude increases with increasing fluence. Therefore, it is appropriate to reduce the energy deposited in the thermoelastic region. It may be reasonable to reduce the spot size of the femtosecond laser so that the size of the nonthermal ablation area is limited. According to Figure 10a, as the spot size decreases, the laser energy is focused on a smaller area. The spot size in the calculation is consistent with that in the literature of femtosecond laser experiments [33,34]. Figure 10b shows the region size of thermoelastic vibration under different spot sizes. Molecular dynamics simulations were conducted under a series of laser fluences, and the maximum and minimum thresholds for thermoelastic vibrations were calculated, with values of 1050 J/m² and 2550 J/m², respectively. Subsequently, the length occupied by the laser fluence within the threshold range was calculated at different spot diameters. It can be seen that with the decrease in spot size, the thermoelastic region is obviously reduced. It can be seen that with the decrease in spot size, the thermoelastic region is obviously reduced. It is worth noting that a small spot size may result in a smaller effective ablation area. Therefore, when a femtosecond laser is used to process a metal surface, it may be feasible to convert a single pulse into multiple pulses with a small spot size.

4. Conclusions

In this paper, the thermoelastic vibration behavior of nickel film irradiated by a femtosecond laser was described in detail by the TTM-MD model. The main conclusions are as follows:

(1)

Stress waves are generated in nickel films under femtosecond laser irradiation. The laser-induced stress wave propagates and reflects continuously in the film, which leads to thermoelastic vibration in the nonthermal ablation region.

(2)

Thermoelastic vibration can be divided into two stages, which are bounded by thermal equilibrium. Before the thermal balance, the deposition of laser energy in the nickel film increases the energy of the collective atomic motion, which leads to continuous expansion behavior. After reaching the thermal balance, the relaxation of laser-induced stress leads to the periodic expansion and contraction of the nickel film, accompanied by changes in structure and temperature.

(3)

The amplitude of thermoelastic vibration increases with increasing fluence, which is the result of the increase in the energy of collective atomic motion. At high fluence, the propagation time of the stress wave is prolonged by a large expansion rate and more intense melting.

(4)

Reducing laser energy deposition in the nonthermal ablation region is an effective way to reduce thermoelastic vibration. This can be achieved by reducing the size of the femtosecond laser spot. When a femtosecond laser is applied to a metal, it may be feasible to transform a single pulse into multiple pulses with a small spot size.

Notably, considering the effectiveness of the TTM in defining the interaction mechanism between femtosecond lasers and materials, particularly metals, various correction methods have been developed to obtain more accurate and reasonable computational results. In terms of material thermal properties, the dependencies of numerous parameters, such as electronic heat capacity and the electron-phonon coupling constant, on factors such as electronic temperature are continuously being corrected, leading to a more precise description of thermodynamic behaviors. Regarding the deposition of laser energy into materials, the dynamic changes of various optical properties with pulse width and laser fluence are carefully taken into account within the model, which facilitates a more precise characterization of the material’s absorption process of laser energy. These endeavors provide solutions for accurately defining numerous physical phenomena of materials under femtosecond laser irradiation, including melting, photomechanical spallation, and phase explosion. Therefore, the further development of a new TTM-MD model considering multiple correction factors to more accurately and in greater detail reveal the mechanism of thermoelastic vibration is the next research direction.