Laser-generated shock wave attenuation aimed at microscale pyrotechnic device design

To meet the rising demand for miniaturizing the pyrotechnic device that consists of donor/acceptor pair separated by a bulkhead or a thin gap, the shock initiation sensitivity in the microscale gap test configuration is investigated. For understanding the shock attenuation within a gap sample (304 stainless steel) thickness of 10∼800 μm, the laser-generated shock wave in water confinement is adopted. The shock properties are obtained from the free surfacevelocity by making use of a velocity interferometer system for any reflector (VISAR). Analytical models for plasma generation in a confined geometry and for evolution and decay of shock waves during the propagation are considered. The shape and amplitude of the laser-driven initial pressure load and its attenuation pattern in the gap are effectively controlled for targeting the microscale propagation distance and subsequent triggering pressure for the acceptor charge. The reported results are important in the precise controlling of the shock strength during the laser initiation of microscale pyrotechnic devices.


I. INTRODUCTION
The performance of a pyrotechnic device that consists of donor/acceptor pair separated by a bulkhead or a metal gap relies on shock attenuation in the metal and shock sensitivity of the energetic materials.When shock wave emerges from a detonating donor, the amount of reflection and transmission of the shock waves at the interfaces of donor/bulkhead and bulkhead/acceptor is determined by the acoustic impedance of these materials.The transmitted shock wave starts to decay along the path of propagation in the bulkhead; if the shock pressure is higher than the initiation pressure of an acceptor, then acceptor is triggered.Thus the shock sensitivity of acceptor must be known along with the critical thickness of bulkhead in an effort to optimize the system. 1,2In order to address the rising demand for miniaturized pyrotechnic devices, the study on shock initiation sensitivity of small scale gap configuration is necessary.The gap thickness less than a millimeter and how the pressure is attenuated through the gap must be understood precisely.
In this research, therefore, a laser-based system is designed to send shock waves into a 304 stainless steel using a Q-switched Nd:YAG laser operating at 532 nm wavelength with pulse energy up to 400 mJ and 5 ns pulse width.To achieve high pressure shock wave for a given energy, laser plasma is generated under a water confinement regime (WCR). 3Free surface velocity of a target is measured by making use of a velocity interferometer system for any reflector (VISAR).By varying the thickness of a target, shock pressure is determined in terms of the distance of wave propagation.Analytical models for plasma generation in a confined geometry and for formation and attenuation of shock waves is also performed to understand the behavior of the shock wave inside of a target.The shape and amplitude of the laser-driven initial pressure load and its attenuation 055314-2 H. Yu and J. J. Yoh AIP Advances 6, 055314 (2016)   pattern in the gap are controlled for targeting the microscale propagation distance and subsequent triggering pressure.Therefore, the miniaturization of pyrotechnic device is investigated from the present laser-based system.

II. EXPERIMENT
Shock waves in orders of GPa are generated by a Q-switch Nd:YAG laser (Spectra, Lutronic Corp.) with 100-400 mJ at 0.532 µm of wavelength and 5 ns of Gaussian pulse.Figure 1 depicts the schematic view of experiment setup and target assembly.The laser beam travels through a Nd:YAG mirror, a focusing lens, water confinement, and arrives at the front side of the target.When a laser beam is irradiated on the target surface, target material within a few skin depth is ablated accompanied by a plasma generation.Since the plasma is confined by a transparent material to the incident laser beam, it induces stronger and longer recoil pressure than that of a direct ablation. 3ere, a tap water layer of 3 mm thickness is used as a confining medium which has approximately 100 % transmission of a driving laser beam at one order of J/cm 2 fluence range. 4304 stainless steel is used as a target material whose thickness is varied from 10 to 800 µm to understand the physics of shock wave according to the distance of wave propagation.A spot diameter of 2.11 mm is obtained by using a plano-convex lens of 100 mm focal length which is chosen relatively long, for reducing the error during the installation of a spot diameter on the sample.The large ratio of spot diameter to thickness of the target and the flat-top spatial profile of a laser beam make it possible to maintain a 1-D planar shockwave for correct determination of a shock state. 5Since the laser used in this experiment was designed for medical use, there is no sync out port to synchronize with the other laboratory apparatus.Instead, a photodetector was used to trigger the oscilloscope by collecting scattered light when the beam is reflected from the mirror.The oscilloscope to record output signals from VISAR is synchronized with the arrival time of shock waves at the back free surface of a target.
VISAR is set up to measure back free surface velocity of a target which starts to move when shock waves arrive.The principle and details of the VISAR used in our system were explained in authors' earlier works. 6The probe beam of the VISAR is precisely aligned to receive reflected beam from the back surface of the target and moved back and forth to obtain the maximum output signals at the exact focal point.The resolutions of time and velocity are respectively set to 1 ns and 1 m/s by installing the delay leg of 200 mm length.The measurement error of velocity is lower than 2 % in the present experimental set up.
From the back free surface velocity, the longitudinal stress (or pressure) of shock wave is determined by the Rankine-Hugoniot jump condition and elastoplastic behavior of the material.The longitudinal stress is determined by the expressions of Eq. ( 1) where the amplitude is compared with the Hugoniot elastic limit (HEL).
Here, HEL marks the transition from a purely elastic state to an elastoplastic state.u p , U s , C el , and C 0 represent the velocities of particle, shock wave, elastic wave, and bulk sound wave, respectively.By matching impedance at the free surface, the resulting particle velocity is approximated as a half of the free surface velocity, u f which is measured by VISAR. 7Shock and particle velocities are assumed to follow a linear relationship for materials over a wide range of pressures as indicated in the experiments. 8ρ 0 , Y 0 , and S represent density before shock compression, compressive yield strength, and characteristic material constant, respectively.Material properties for calculation of the longitudinal shock stress in 304 stainless steel are as follows: ρ 0 = 7900 kg/m 3 , C 0 = 4570 m/s, C el = 5770 m/s, Y 0 = 205 MPa, S = 1.49. 5,955314-3 H. Yu and J. J. Yoh AIP Advances 6, 055314 (2016)

III. ANALYSIS
An analytical method is developed to predict the shock behavior in the steel, such as evolution and attenuation of a shock front during its propagation.The method outlined here provides pressure, pressure gradient, velocity, acceleration, and position of a shock front. 10,11 of evolution and attenuation of shock waves are explained from the hydrodynamic point of view, 12 such that mass and momentum are conserved and the coalescence of compression waves form shock waves.Shock waves evolve and decay by a sequence of compression and release waves.Also the perturbation waves are generated by the infinitesimal change in the pressure at the tail of a propagating pressure pulse.In the laboratory frame, perturbation waves immediately behind the shock front propagate at the velocity of C + u p .Here, C and u p represent sound velocity and particle velocity of each preceding shock front at a pressure, P. Since shock velocity is lower than the sum of sound and particle velocities, pressure of the shock front increases or decreases when perturbation waves overtake the shock front.Finally, a leading edge of the shock front becomes steep until the shock front reaches the maximum pressure, and then decays according to the profile of input pressure pulse during the propagation.Here, the energy losses by viscosity, thermal conduction, and reflection of perturbation waves at the shock front are not taken into account in the model. 8,12ased on the assumptions of shock discontinuity and isentropic flow, an expression for the shock evolution and attenuation is known from the conservation laws of mass and momentum. 13hen we can obtain the following expression for a 1-D planar geometry, where x is the space coordinate, dP/dx is the variation of shock pressure, and ∂P/∂ x is the pressure gradient immediately behind the shock front.In order to solve Eq. ( 4), U s , u p , C, and ∂P/∂ x are expressed in terms of pressure.The conservation laws of mass and momentum for a steady 1-D planar shock wave are From Eqs. ( 3) and (6), Here, the sound velocity is approximated along the Hugoniot, Eq. ( 10) was derived from the U s -u p linear relation, Eq. ( 3) and standard assumption of Gruneisen parameter, Eq (11). 5 Γ 0 and Γ are the Gruneisen parameters for the initial density and the shocked state, respectively.The Mie-Gruneisen parameters of 304 stainless steel and a water are provided in Table .I.An expression for ∂P/∂ x at specific position was obtained in the following way. 10Pressure waves of different magnitude are emitted from the interface between target and confining medium at different times during the plasma generation, and then transmitted to the target material at a different speed which is a sum of the particle and sound velocities.Due to the difference in the departure time of pressure waves at the interface and the difference of particle and sound velocities depending on pressure amplitude, the arrival time of pressure waves reaching a specific position x is changed with the distance of wave propagation.In all earlier works, the ablation surface or the interface between target and confinement was fixed at the reference position (x = 0) without considering the particle velocity.The temporal profile of an input loading was simplified as a triangular pulse which is linear in time.However, in the present work, the displacement of the ablation surface and the actual temporal profile of input pressure loading are added to the consideration for more realistic results.
Since the profile of plasma pressure is expressed as a function of time, the pressure derivative with respect to time ∂P/∂t must be obtained prior to expressing in the form of ∂P/∂ x.
Here x A is the position of the ablation surface and ∂P/∂t| x A is the time derivative of pressure at the ablation surface.x A and dC/dP are expressed as follows. where Since the velocity of pressure wave is C + u p , ∂P/∂ x becomes The variation of shock pressure ∆P according to propagation distance ∆x is calculated by using Eq. ( 4) with above variables written in terms of pressure.In return the shock pressure in x P (x) is obtained by the iteration, H. Yu and J. J. Yoh AIP Advances 6, 055314 (2016) The acceleration and time of the shock front at position x are computed from To simulate the temporal profile of the plasma pressure in the confined geometry according to temporal shape of the laser pulse, 1-D analytical model was applied. 14When a laser beam is irradiated at the interface between a target and a confinement, rapid generation and expansion of plasma induce strong shock waves which propagate into the two mediums.The fluid motion behind the shock waves results in a displacement of the interface.Using the Hugoniot relation, particle velocities and the time derivative of the interface thickness at time t can be related as follows.
Here, shock impedance of materials, Z i = ρ 0,i U s,i can be regarded as acoustic impedance for a pressure range of the experimental condition.The subscripts 1 and 2 represent properties for target and confining mediums, respectively.The deposited laser energy during time interval I (t) dt is used as the pressure-volume work P (t) dL and the internal energy of the plasma E i (t) dL.The fraction α and 1 − α of the internal energy are respectively used for the thermal energy and ionization of the gas.By assuming the plasma as an ideal monatomic gas, the pressure is represented with the thermal energy and corrective factor α. 15 Therefore the conservation of energy can be expressed as Here P p (t), L (t), and E i (t) are the pressure, thickness, and internal energy density of the plasma, respectively.E T (t), I (t), and α represent the thermal energy density of the gas, the absorbed laser intensity, and the ratio of thermal to internal energy density, respectively.The pressure, impedance, and laser intensity are expressed in the unit of Pa, kg/m 2 s, and W/m 2 , respectively.By combining Eqs. ( 20) and (21), it becomes The temporal profile of plasma pressure is obtained by solving the 2 nd order ordinary differential equation.
Since this model assumes that the plasma pressure is strong enough to send shock waves into the adjacent two materials, the model should be used with care only when target material is ablated.Therefore we considered the ablation threshold in conjunction with the laser fluence in time in order to find the initiation point of the ablation.For a Gaussian laser pulse which is used in the experiment, the intensity profile and laser fluence in time are expressed as follows, Here F, F th , and σ are the total laser fluence, ablation threshold of a target material, and standard deviation of Gaussian distribution, respectively.Consequently, the generation point of the plasma and subsequent shock waves is shifted by βσ.By assuming that the plasma experiences adiabatic cooling process after the pulse duration of the laser beam, the plasma pressure is expressed by the following relation.
where τ and λ represent the pulse duration of the laser beam and the ratio of specific heat, respectively.

IV. RESULTS AND DISCUSSION
The free surface velocity for various thicknesses was measured by VISAR to investigate the shock attenuation with distance of the wave propagation.Figure 2 shows the velocity profile of the free surface for various thicknesses of a target at the laser pulse of 8.6 J/cm 2 .The first and the subsequent peaks are caused by the arrival of the shock waves at the free surface and its reverberations, respectively.From the amplitude of the first peak, shock pressure is determined.As the thickness of the target decreases, the time interval and the fluctuation amplitude of the free surface velocity between each crest and trough become shorter and narrower.In the case of a thin foil target whose double transit time of a shock wave is much shorter than the duration of the plasma pressure loading, numerous reverberations of shock and release waves rapidly accelerate a target without noticeable fluctuations. 15This phenomenon shows the experimental difficulty where the measurement of the first peak by a single shock is a challenging task for a thin target below 100 µm thickness.Therefore, the pulse width and temporal profile of the input pressure loading are quite essential when determining the minimum thickness limit for a potential miniaturization of a such system.The analytical model for laser-induced plasma and evolution/attenuation of shock is applied near the ablation surface where the accurate measurement of the shock properties is not possible due to the minimum thickness limit.The pressure and particle velocity of a shock front obtained both numerically and experimentally are represented in terms of the distance of wave propagation in Fig. 3. Solid lines denote the theoretical results based on the hydrodynamic analysis.Dots are experimental results shown with maximum and minimum deviations.Here, the calculation is validated by showing a good agreement between the calculation and experiment.The shock 055314-9 H. Yu and J. J. Yoh AIP Advances 6, 055314 ( behavior such as how its front is evolved and attenuated within the thickness limit shows a different characteristics with respect to a laser fluence.In the case of 10.9 J/cm 2 , the shock front evolves until reaching a maximum value and then starts to decay.Whereas the shock front is immediately attenuated without evolution in the beginning for all other low fluence cases.This shows that the distance to shock steepening is necessary to estimate the exact magnitude of shock pressure at a desired distance.Therefore, such distance must be considered in addition to the minimum thickness limit. The different behavior of shock waves is resulted from the reflectivity and ablation threshold of a target.Generally, total reflectivity of metals above the ablation threshold starts to decrease rapidly as laser fluence is increased and kept at constantly rising.In the case of a nanosecond Nd:YAG laser ablation at room temperature and pressure, the total reflectivity of metals starts to decrease in the fluence proximity of 1∼12 J/cm 2 . 16Since the fluence range of our experimental condition is within the transient region, the reflectivity of the target surface must be considered for exact plasma pressure from which exact shock behavior can be obtained.Therefore, Eq. ( 22) is corrected by adding the reflectivity term to consider the change of the reflectivity according to laser fluence, such that The plasma pressures generated by a Gaussian laser pulse at various fluences are represented in Fig. 4(a).As the laser fluence increases, the slope of the pressure gradually increases with a similar rise time to a maximum value for all cases, then the pressure starts to decay slowly.Figure 4(b) shows the temporal profile of laser intensity and plasma pressure which are normalized by their maximum values without considering an ablation threshold.Since the plasma pressure is mostly influenced by the absorbed laser intensity which is used to heat and vaporize the target, the temporal profile of the plasma pressure behaves similarly to the temporal profile of the laser intensity until reaching the peak.Whereas the plasma pressure smoothly decays because it is controlled by the rate of heat conduction from the metal and vaporization to the colder enclosed medium and pressure-volume work done on the neighboring medium. 17n the analytical model, laser-induced shock waves are generated when a deposited laser fluence reaches the ablation threshold of a target surface.Therefore ablation threshold and deposited laser fluence in time have an influence on the amplitude and initiation time of shock wave, which are calculated by using Eqs.( 23) and (24).Figure 5(a) shows intensity profile of a Gaussian laser pulse according to the laser fluence and ablation threshold which is represented as an area filled with color and shades.As the laser fluence increases, the peak amplitude of laser intensity rises due to the same standard deviation of a Gaussian distribution, and consequently the occurrence time of laser ablation goes forward.This means that different magnitude and temporal shape of the plasma pressure is directed to the target surface, resulting in the different behavior of the shock waves in the medium.As shown in Fig. 5(b), the occurrence time of plasma pressure is the same with that of the laser ablation, and the plasma pressure starts to follow the previously calculated results after the onset of plasma generation.The estimated reflectivity of 304 stainless steel and the time occurrence of laser ablation are represented in Table .II.
Using the analytical model validated from the experimental results, it is possible to simulate the laser-induced plasma pressure and the subsequent shock waves in the suggested thin medium.Figure 6 represents pressure, velocity, and acceleration of a shock front as a function of distance of wave propagation, respectively.As shown in Fig. 6, the overall shape of the shock pressure and velocity is similar to the temporal profile of plasma pressure toward a target surface that was shown in Fig. 4(a) and 5(b).This means that evolution and attenuation of a shock front is greatly influenced by the temporal shape of an input pressure or a laser pulse, and the position of shock front is affected by the material properties of a target and magnitude of shock pressure.This result is consistent with the existing reports where input pressure pulse in a triangular or trapezoidal shape causes the shock front to evolve and decay in a triangular or trapezoidal form. 10,11  When pressure waves exceeding HEL is transmitted through the elastoplastic material, the waves are divided into an elastic precursor of HEL amplitude and a plastic shock wave.The measurement of HEL is necessary as it is related to the mechanical properties of the target.The strain rate influences the value of HEL which is much higher in the laser-driven shock experiment when compared with gas guns.Therefore it is essential to measure HEL in the laser-driven shock system as discussed here.The only way to determine HEL is by distinguishing the elastic-plastic inflection from the free surface velocity. 18Shown in Fig. 2, 304 stainless steel has an indistinguishable HEL point that is different from those of 316L and 55C1 steels. 19Therefore, what we proposed in this analysis favorably determines the inflection point by drawing an acceleration curve along with the free surface velocity as represented in Fig. 7.During the period in which the velocity increases, the acceleration curve has two peaks.Theses waves are caused by the arrival of an elastic precursor and a plastic shock wave, respectively.Thus, it is possible to determine HEL even if material has a blurred distinction.The amplitude of HEL is calculated from the conservation of momentum, from which the dynamic yield strength is obtained. 7L = ρ 0 C el u p,HEL (28) Here ν and σ y dyn represent the anisotropy coefficient and the compressive yield strength at high strain rate of a material, respectively.We have the following values on 304 stainless steel: u f ,HEL = 23 m/s, HEL=524 MPa, σ y dyn =330 MPa, which remain almost constant regardless of propagation depth all within a few hundred orders of a micron.The dynamic yield strength is different with the quasi static yield strength σ y = 205 MPa.This means that the compressive yield strength is dependent on the strain rate. 20= 1 C 0 du p dt (30) The yield strength at the strain rate 10 5 s −1 is approximately 1.6 times higher than the yield strength at the quasi strain rate 10 3 s −1 .Therefore the amplitude of an elastic precursor and the dynamic yield strength must be considered for the present setup, aimed at studying the shock attenuation in a miniaturized pyrotechnic unit.

V. CONCLUSION
In this research, some preliminary analysis of shock behavior within a microscale steel sample was performed in an effort to collect necessary data for the miniaturization of a laser-based pyrotechnic device that works in a standard donor/acceptor configuration.The experiment and theoretical calculations are conducted to figure out the relation between the input pressure and its attenuation.The minimum sample thickness required for sharpening the shock front and for avoiding pressure decrease by multiple reverberations of shock and release waves were determined, in order to obtain exact pressure at a desired distance.A method of determining the HEL for such microscale sample was also developed.Thus the reported results can be used in the precise controlling of the shock strength during the laser initiation of microscale pyrotechnic devices.

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FIG.2.Temporal profile of the free surface velocity according to the thickness of a target at laser energy of 8.6 J/cm 2 .

FIG. 3 .
FIG. 3. Evolution and decay of shock pressure represented by analytical model and experimental results.

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FIG. 4. Temporal profile of (a) Plasma pressures at laser pulse of various fluence and (b) Normalized laser intensity and plasma pressure for 10.9 J/cm 2 .
Figure 6 also describes the acceleration of a shock front with the propagation distance.

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FIG. 5. (a) Laser energy used for ablation of 304 stainless steel and (b) Temporal profile of plasma pressure after laser ablation according to laser fluence.

FIG. 6 .FIG. 7 .
FIG. 6.(a) Pressure, (b) velocity, and (c) acceleration of a shock front with respect to the distance of wave propagation at various laser fluence.
The mechanism

TABLE I .
Mie-Gruneisen parameter of 304 stainless steel and water.

TABLE II .
Estimated reflectivity of target surface and time occurrence of laser ablation according to laser fluence.