Molecular Dynamics Study on Wavefront Structure of Single Crystal Copper under Femtosecond Laser Loading

LIU ChengLEI Jiehong

Physics and Space Science College，China West Normal University，Nanchong 637001，China

Abstract：We used computer simulation with the advantage of easily designing the material structure to design single crystal copper samples，studied the mechanical behaviors of this material under different shock compression conditions，and comparatively analyzed the reasons for the formation of different mechanical behaviors. An important macro performance of metal mechanical behaviors under shock compression is the shock wave front. In fact，the structure and the evolution of the shock wave front during the plastic phase are determined by its microscopic process，i.e.，the development process of shear flow and energy dissipation in the plastic phase. Due to the limitation of the resolution of experimental instruments，it is not yet possible to obtain a clear shock wave front structure through experiments. By means of molecular dynamics (MD) simulation，we obtain the velocity and coordinate information of every atom in each system，then compare the plastic deformation behavior of the material under different impact conditions，and finally obtain the impact of the shock conditions on the structure of the shock wave front.

Key words:single crystal copper；mechanics；plasticity；shock wave

Introduction

As we all know，the factors that affect the plastic deformation behavior of materials can be divided into external factors and internal factors.The external factors are mainly the loading method，temperature and other conditions，while the internal factors include the intrinsic properties，structural characteristics and size effects determined by the interaction between atoms.In recent years，metallic materials have been extensively studied because of their unique mechanical behaviors compared to traditional coarse-grained materials，such as high strength and high ductility[1-3].

Understanding the structure of the shock wave front is crucial for understanding the dynamic response characteristics of the material under shock compression.Researchers[4-7]have summarized the relationships between the rise time of the shock wave and the shock pressure in different solid materials from a large number of experimental results.It is believed that the strain rate is proportional to the fourth power of the shock pressure，and the rise time of the shock wave is inversely proportional to the shock pressure.Its theoretical basis is explained from the viscosity theory and the universality of this theory has been extended recently[8-9].However，this theory only provides the outline of the shock wave front (i.e.，the width) and does not give the microscopic mechanism that causes this change.Due to the limitation of the technical level of modern measurement systems，the structure of the shock wave front can only be guessed，it is very important to use the molecular dynamics (MD) method to interpret the physical nature of the above experimental measurement data and construct the corresponding physical model.

With the applications of the MD method and the Monte Carlo (MC) method in the study of shock wave，more and more complex internal structures of shock wave front have been revealed in terms of atom and mesoscale[10-15].

1 Sample Construction and Simulation Method

For the process of the femtosecond laser driving the shock wave in the metal material，after the femtosecond laser action ends and after the electron ions relax to reach equilibrium，the MD method is used to model and calculate the expansion process of the plasma and the propagation process of driving shock wave in target.Initial configuration is set to ideal single crystal copper (about 600 nm) composed of 30×30×300 fcc unit cells with a total of 1.08 million atoms.Thex，yandzaxes are taken along the[100]，[010] and[001] crystal directions，respectively.The periodic boundary conditions are applied along thexandydirections，i.e.，if there are particles running out of the model，particles in the same state must return to the model from the opposite interface to ensure that the number of particles in the simulation system is constant.In addition，the method of nearest mirror image is taken to calculate the force between atoms，thereby eliminating the boundary effect of the atomic forces at the boundary.A free boundary without applying any external constraints is set along thezdirection.The initial temperature of the sample is set to 5 K,the lattice constantais 0.365 nm，and then the simulated sample is allowed to fully relax under the NVT ensemble (10 ps).The atomic motion equation is integrated using the velocity Verlet algorithm，and the time step is taken as 0.001 ps.In order to simulate the shock wave within a given time，the entire simulation system is divided into many areas with a certain thickness along the impact direction (the width of each area is 3a).The physical quantities such as the pressure in each region and the symmetry parameters of the atom center，are calculated.

The program is written according to the different formats of the dual temperature model.The ablation process of the single crystal copper is calculated under the condition of femtosecond laser pulse with a femtosecond laser pulse at a wavelength of 800 ns，a pulse width of 35 fs，and a laser power density of 2.55×1020W/cm2.The calculated temperature when the electron temperature and ion temperature on the surface of the copper material reach equilibrium are 7.75×105K and 7.80×106K，respectively.

2 Simulation Results and Discussion

2.1 Propagation process of shock wave in single crystal copper

In order to understand the propagation process of shock waves in single crystal copper，we used simulation calculations to load a femtosecond laser pulse at a wavelength of 800 ns，a pulse width of 35 fs，and a laser power density of 2.55×1020W/cm2on the single crystal copper.The relationship between the free surface velocity which corresponding to shock wave propagation in the target and time is shown in Fig.1.The compression waves generated by the shock move towards the free surface of the target and the flying disc，respectively.When the compression wave reaches the free surface of the target，the reflection is a sparse wave.It can be seen from Fig.1(a) that the impact reaches the free surface of the target at 15 ps and this time point corresponds to the pointAin Fig.2.Figure 1(b) shows the propagation process of the sparse wave reflected from the free surface of the flying disc and the target.The sparse wave interacts at 29 ps and this time point corresponds to the pointBin Fig.2.The sparse wave interaction makes the tensile stress appear in the target，which propagates towards the free surface of the target as shown in Fig.1(c).It reaches the free surface at 40 ps and the speed of the free surface begins to decrease，corresponding to the pointCin Fig.2.With the time increases to 45 ps (the pointDin Fig.2)，two peaks appear in the tensile stress in Fig.1(d) and the void simultaneously begins to appear.The pointEin Fig.2 corresponds to the spallation of the material.The interaction of two series of waves corresponds to the two peaks in Fig.1(d) due to nucleation and the growth of the cavity，resulting in different amplitudes of waves in the free surface velocity map (pointsFandGin Fig.2).

Fig.1 Propagation process of shock wave in target with a pulse at (a) 5，10 and 15 ps；(b) 16，21，25 and 29 ps；(c) 34，36 and 40 ps；(d) 41，43 and 45 ps

Fig.2 Free surface velocity versus with time

Figure 3 shows that the atomic central symmetry parameter (CSP) is a function of time when the laser power densityIis 2.55×1020W/cm2and 2.55×1022W/cm2，respectively.It can be seen that the crystal structures are all complete lattices att=0.As the impact time increases，the mass fraction of the complete lattice decreases，and the stacking fault structure begins to appear in the crystal.When the laser power density is 2.55×1020W/cm2，stacking faults and defects begin to appear at about 3 ps.With the further increase of time，the mass fraction of stacking faults and defects increases.It can be seen from Fig.3 that the increase in the speed of the flying disc results in a faster increase in the mass fraction of stacking faults and disloctions.

Fig.3 Atomic CSP as a function of time：(a) I=2.55×1020 W/cm2；(b) I=2.55×1022 W/cm2

2.2 Plastic deformation process under shock compression

We simulated the plastic deformation process under shock compression when a single crystal copper was loaded with a femtosecond laser pulse at a wavelength of 800 ns，a pulse width of 35 fs，and a laser power density of 2.55×1020W/cm2.The impact pressure and particle velocity profiles are obtained from the statistical average of the velocity and stress values in three lattice constant (about 10.08 nm) long boxes along the impact direction.Figure 4 shows the stress and the particle velocity profiles of the single crystal copper，respectively.It can be seen from Fig.4(a) that the Hugoniot elastic limit (HEL) in shock wave physics is 47 GPa，which is the elastoplastic transition point of materials in response to shock compression.

Fig.4 Simulation result：(a) stress distribution under shock compression；(b) particle velocity distribution

We define the distance of the impact stress or particle velocity from 0 to the highest 95% (post-shock plateau value) as the width of the shock wave front，and it is marked with black lines and arrows in Fig.4(b).The impact stress profile and the particle velocity profile shown in Fig.4 both reflect the elastoplastic double-wave structure of the weak shock wave.In Fig.4，the slope of the waveform curve in the elastic phase is very large，and the stress rises substantially linearly，indicating a quick rise in the elastic precursor wave.At the same time，it can also be seen that as the impact time and the distance increase，the width of the shock wave front at 14 ps is greater than that at 10 ps.The reason is that the velocity of the elastic wave under weak shock loading is higher than that of the plastic wave，which causes the shock wave front to widen with the impact distance or time.Similar phenomenon was observed in the experiment of shock wave generated by a laser loading coarse-grained aluminum film[16-17].Thus the shock wave obtained by MD simulation can be recognized to be stable based on the similarity of the shock wave profiles at different moments.

Figure 4 shows that there are still some fluctuations in the stress in the post shocked area，which indicates that there are some local plastic deformation processes in the rear area，but they cannot be reflected on the impact stress profile as a whole(dsis the width of the shock wave front).We roughly attribute this area which basically reaches stability to the plastic completion stage.

Figure 5 is the diagram of the deformed atomic structure after the shock compression of 10 ps when the single crystal copper is loaded by the femtosecond laser at laser power densities of 2.55×1020W/cm2and 2.55×1022W/cm2，respectively.It can be seen that a small amount of dislocations occurs after the shock wave.This shows that the plastic deformation mechanism of the single crystal copper in response to shock compression is mainly the emission and the propagation of dislocations.In Fig.5，as the laser power density increases，the dislocations of the single crystal copper increase and the impact stress and strain also increase，which is consistent with the basic understanding in shock wave physics.

Fig.5 Deformed atomic structure of single crystal copper at a certain laser power density I：(a) I=2.55×1020 W/cm2；(b) I=2.55×1022 W/cm2

Summarizing the above analysis，we can divide the shock wave propagation process in the single crystal copper into three stages，namely the elastic stage，the plastic stage and the plastic completion stage.The main concern in this work is the plastic stage dominated by dislocations.During shock compression，elastic deformation occurs firstly in the area where the shock wave reaches and forms a relatively narrow elastic precursor wave propagation area.Then as the stress increases，plastic deformation occurs，forming a plastic wave area with a certain width.Finally，as the stress further increases，the dislocations start to emit，forming a sufficient plastic flow and allowing the shock wave to reach a stable stress state.

Figure 6 shows the impact stress and particle velocity profiles of the single crystal copper at different impact strengths.It is easy to find that as the femtosecond laser power density and the impact stress increase，the shock wave front becomes steeper and steeper,and the width decreases.It means that as the impact stress increases，the time required for plastic deformation to form is shorter，and it is easier to reach a stable post-shock equilibrium state，which is consistent with the experimental results and our intuitive understanding of shock wave.

Fig.6 Profiles of single crystal copper after 10 ps impact compression:(a) stress profile;(b) particle velocity profile

From Fig.7，we can see that the calculated plastic deformation strain rate of the single crystal copper is proportional to the first power of the impact stress，which is quite different from the experimental fourth power law[14].The reason is that the early impact loading experiments are carried out on light gas guns，and the resulting strain rates are all below 108 s-1，while the strain rates in MD simulation are all above 109 s-1(corresponding to the experiment of shock wave generated by laser loading).It is unknown whether the fourth power law summarized by Grady can be applied under high strain rate conditions because the physical mechanism of dynamic response at high strain rates is different from that at low strain rates.It can be seen from the fitted data that at high strain rates，the strain rate increases，the stress increases faster，and plastic deformation and failure are more likely to occur.

Fig.7 Comparison of calculated stress-strain rate relationship with Grady’s empirical relationship

3 Conclusions

The factors that affect the plastic mechanism and the structure of the shock wave front are preliminarily discussed.The research results show that the shock wave front under femtosecond laser shock compression is a macroscopic manifestation of the microscopic plastic deformation process.When the shock wave just sweeps through the single crystal copper，the shock wave dislocations nucleate and propagate into the crytal grain after the single crystal copper undergoes a brief elastic deformation.The macroscopic plastic deformation completes and a basically stable post-wave state is formed after the shock wave front.

By observing the width of the wave front under different loading conditions，it can be found that for single crystal metals，when the pulse width of the femtosecond laser is fixed，the width of the shock wave front decreases with the increase of the power density of the femtosecond laser，which is the same as traditional non-nano materials.However，we find that due to the timing of the plasticity of dislocations on the wave front，the width of the wave front will increase.This is the characteristic of metals in response to shock loading.

Finally，the relationship between the stress and strain rates of the single crystal copper at different laser power densities is quantitatively studied，and we find that there is a big difference with the previous experiences summarized from the experiments.We believe that the reason for these different relationships is that at high strain rates，the physical mechanism of dynamic response is different from that of low strain rates.