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Induced evaporation of metal from an aluminum surface by a normal pulse neodymium laser

Johnson, Christopher Brinton

Monterey, California. Naval Postgraduate School

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411 Dyer Road / 1 University Circle Monterey, California USA 93943


Christopher Brinton Johnson


Monterey, California




Christopher Brinton Johnson

September 1979

Thesis Advisor: F. Schwirzke

Approved for public release; distribution unlimited.

6 fA § ~~









S$. TYPE OF REPORT &@ PERIOO COVERED Master's Thesis; September 1979


. TITLE (and Subtitte) Induced Evaporation of Metal From an Aluminum Surface by a Normal Pulse

Neodymium Laser


Christopher Brinton Johnson


Naval Postgraduate School


Monterey, California 93940


September 1979

. MONITORING AGENCY NAME & ADORESS(i/ different tram Controiiing Olltce) | 18. SECURITY CLASS. (of (Ate report)


Naval Postgraduate School Monterey, California 93940



Approved for public release; distribution unlimited.

- DISTRIBUTION STATEMENT (co! the sbetract entered in Biock 20, ti dilierant from Report)


. KEY WORDS (Continue on reverse aide if neceseary and identity by biock number)

Laser Induced Evaporation Neodymium Laser

20. ABSTRACT (Continue an reveree side if necessary and identify sy biock number)

Laser induced evaporation of material from the surface of an aluminum target in a vacuum was studied. Based on a literature examination, material removal using a normal pulse laser was judged to be more efficient than for a Q-switched laser. The experiment vas conducted using a neodymium glass laser modified for normal 3 pulse operation. The energy density was varied from 8.5xl02 J/cm here no breakdown occurred to 5x10 T/om* where the threshold for

DD ons; 1473 cvition oF | nov 6818 omsoLeTe

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breakdown was exceeded. The normal pulse duration was 600 us. Analysis of the ejected material was achieved by using a Hughes Ionization Gauge placed in the path of the ejected Material. Oscilloscope traces of the ionization gauge output Show that the gauge "Sees" what is flying past it. There

1S good correlation between laser radiation, plasma radiation and ionization gauge fluctuations. The ionization gauge

gave distinguishable signals for ions, electrons, and

neutral particles ejected from the target surface. Signal sequence was dependent on the particle velocity. By measuring the elapsed time after ejection from the surface and the target to collector distance, the first arriving neutral particle velocity was determined to be 5.2x.104 cm/s.

DD Form. 1473 qj dang ___UNCLASSIFIED S/N 0102-014-6601 2 SECUMITY CLASSIFICATION OF THIS PAGE(PREM Date Entered)

Approved for public release; distribution unlimited.

Induced Evaporation of Metal from an Aluminum Surface by a Normal Pulse Neodymium Laser


Christopher Brinton Johnson Maso~, United States Army B.A., Washington State University, 1968 M.S. University Of “Southern Calafornia, 1975

Submitted in partial fulfillment of the requirements for the degree of


Erom une



Laser induced evaporation of material from the surface of an aluminum target in a vacuum was studied. Based on a literature examination, material removal uSing a normal pulse laser was judged to be more efficient than for a Q-switched laser. The experiment was conducted using a neodymium glass laser modified for normal pulSe operation.

: J/om* where no

The energy density was varied from 8.5xl0 breakdown occurred to 5x10° 3/em- where the threshold for breakdown was exceeded. The normal pulse duration was

600 us. Analysis of the ejected material was achieved by using a Hughes Ionization Gauge placed in the path of the ejected material. Oscilloscope traces of the ionization gauge output show that the gauge "Sees" what is flying past it. There is good correlation between laser radiation, plasma radiation and ionization gauge fluctuations. The 1onization gauge gave distinguishable signals for ions, electrons, and neutral particles ejected from the target surface. Signal sequence was dependent on the particle velocity. By measuring the elapsed time after ejection from the surface and the target to collector distance, the first arriving neutral particle velocity was determined to

be oe cm/s.


ive INTRODUCTION ----- 2-9-2 ------ -- - - - - - - - - - - - - - - - - -- jGdee THEORY 2-22-92 oo a nr a a rn - - - - - - A. REFLECTIVITY ---------------------------------

B. HEATING OF THE MATERIAL ----------------------

C. MATERIAL REMOVAL --------99-------------------

III. EXPERIMENTAL DESIGN 7-992-292-9999 eo A. EQUIPMENT ------------------------------------

1. Laser System -----------------------------

2. Target Chamber ---------------------------

3. Ionization Gauge -------------------------

4, Instrumentation --------------------------

B. PROCEDURE ------------------------------------

1. Ion Gauge MeaSurement Study --------------

2. Material Mass Study ----------------------

IV. RESULTS AND DISCUSSION --------------------------- A. PLASMA ---------------------- = = - - - = - - - - - - - =

B. DESORBED GASES ------------ 2-2-2222

C. NEUTRAL PARTICLES ----------------------------

D. MASS OF MATERIAL REMOVED ---------------------

ie CONCLUSIONS eer re ee ee re ee ee Se BIBLIOGRAPHY -------------+------------+--— ------ --+--- +--+ INITIAL DISTRIBUTION LIST -------- oe ee ere




I wish to thank Robert Sanders, Technician, for his valuable assistance in calibrating and repairing the equip- ment necessary for this experiment. I would like to extend my Sincere gratitude to Professor Schwirzke for his help and guidance throughout the experiment. His comments, sugges- tions, and academic assistance, during the writing of the thesis, were of great benefit. This work was supported by

the Defense Nuclear Agency, MIPR Number 79-512.


A wire explosion 1S a complex phenomenon resulting from very rapid electrical heating of a piece of metal which has a small cross sectional area. The wire 1s heated and under- goes phase change. There is a bright flash of light and a loud report (in air). Thus, the term "exploding wire" has been coined to describe this phenomenon. More appropriately, the term "exploding conductor" will be used to denote the explosion of a single wire, a multiple wire array, a metal foil, or a gas puff by a high current pulse. This technique 1s a relatively simple way of generating a plasma, which can be used to study the dense plasma itself or for further experimental work such as spectroscopy.

The idea of exploding conductors is not a new one [3]. As early as 1773, Edward Nairne conducted experiments exploding a .15 mils diameter iron wire in proving that current in all parts of a series circuit is the same [45]. Eighty years later, Michael Faraday reported on wire explo- sions used to produce a metal film or mirror. The use of exploding wires was of little importance until the work of John A. Anderson in the 1920's. At Mt. Wilson Observatory, he showed that the temperatures involved approached those of the sun, in excess of 3000°C [48]. In the years that followed, research on exploding conductors was principally

conducted by Nabaoka, Kleen, Wrana, Eiselt, Conn, Kvartskhava

and Lebedev [3]. Since 1950, exploding conductors have become a matter of great scientific interest. Perhaps the largest number of researchers to use exploding conductors have capitalized on the extremely short, intense light out- put. These intense pulses of light have been used in spec- troscopy and high speed photography, as well as in radiation chemistry and laser technology as an excitation source for coherent radiation [50].

Other uses of exploding conductors are the production and study of aerosols, production of thin metal films [27], Sllvering of mirrors, and chemical synthesis [1]. In the past 10-12 years, the use of exploding conductors as bridge- type electrical detonators has found widespread application both as a method for multi-point detonation of explosives [46] and as a method to generate shock waves in different materials [9]. Another area of intense research, and one in which this paper is related to, is the use of exploding conductors in simulating nuclear weapons effects. In recent years, progress has been made in the development of exploding conductors as x-ray (photon) sources for use in Simulating nuclear weapons effects on various satellite "black boxes".

The emission of electromagnetic radiation is a result Of suddenly electrically heating thin wires. Four confer- ences held between 1958 and 1967 dealt exclusively with this subject [17,18,19,20]. Toward this end, several varia- tions of the exploding conductor approach to high energy

photon generation have been explored in the past few years.

Single and multiple wire arrays [8,47], cylindrical foils [60], and gas puffs have all been tried to date with limited success. The energy of the emitted x-rays has been too low to be used in nuclear explosion simulations. The problem centers around insufficient energy being coupled to the plasma to allow for radiation in the x-ray region.

Whatever the application or shapes of the conductor, the electrical circuit and the components are similar and perform similar functions, the only difference being in Size and characteristics. Each circuit contains an energy storage device, a switch or triggering device, and an explo- Slve conductor. The energy storage device may be a bank of large capacitors or special multiterawatt generators, such as Pithon at Physics International, Gamble II at the Naval Research Laboratory, or Blackjack IV at Maxwell Laboratories, Inc., [63]. These devices are capable of producing 100 KJ pulses of 50-75 nanosecond duration. This high rate of energy release requires special switching devices and transmission lines. Basically, a Marx-capacitor bank (200~300 KJ) is used to charge a single, large, water capacitor. This capacitor discharges into a water dielec- tric coaxial transmission line that tapers to a 1-22 output resistance at the point where the pulse is extracted into a vacuum diode. Thin wires, ribbons or foils are stretched between the electrode gap in the diode [41]. In the case

Of a wire (13,29,52,53,68] or a foil [66], the initial part

of the electrical discharge causes the conductor to rapidly vaporize uniformly along its entire length. The plasma so formed is then compressed ina Z-pinch. As a result, the plasma consists of "pinched" and "flared" structures char- acteristic of the m = 0 sausage instability to which Z- pinches are susceptible. It is from these pinched and flared areas that radiation is emitted [22]. Gas puff experiments conducted at Physics International are similar to the exploding conductor ones just described except that a gas puff is substituted for the wires. In addition, the gas puff is ionized with microwaves prior to the arrival of the electrical pulse to insure conductivity of the gas.

Most of the methods giving rise to electromagnetic radiation emitted by plasmas are atomic in nature, that is, they are due to transitions involving bound or free states of atoms or ions. There are three basic methods of energy radiation: bremsstrahlung, recombination radiation, and discrete radiation. Bremsstrahlung or free-free radiation occurs when a free electron collides with an ion or neutral particle and makes a transition to another free state of lower energy with the emission of a photon. The energy of the photon may amount to any fraction of the initial kinetic energy of the electron, KT therefore, the bremsstrahlung has a continuum of frequencies determined by KT: At 4

T = 10

Fs °K, the bremsstrahlung lies in the visible and

infrared regions, while at T = LO °K most of the radiation

lies in the x-ray region. Recombination occurs when a


free electron 1S captured by an n-times 10nized atom and makes a transition to a bound state of the (n-1) times ionized atom. The surplus energy 1S emitted as recombina- tion or free-bound radiation. The energy 1s equal to the sum of the kinetic energy of the free electron, KT, and its binding energy. Since free electrons have a continuous energy spectrum, the photons emitted form a continuous energy spectrum. Discrete radiation or bound-bound radiation occurs when a bound electron of an atom or ion 1s excited to a higher energy state by particle collisions. When the electron returns to a lower state, it yields energy at various discrete frequencies. Whenever a plasma contains ions that have not been completely stripped of orbital electrons, this radiation will appear. Generally, it takes greater plasma energy to completely strip atoms or molecules with higher atomic numbers. As the electron temperature is increased more tightly bound electrons are removed. This will be associated with an increase in the excitation energy of the ions. As a result, the line spectrum will shift from the visible to the x-ray region. Although the radiation emitted from exploding conductors is a combination of the above processes, it is dominated at high temperatures by bremsstrah- lung and discrete radiation in the x-ray region [30,34].

The spectral yields and hardness of the emitted radiation from Expileding conductors depends on the ionic state and the

electron temperature which in turn are influenced by a


combination of competing physical processes. Principal among these are the available generator power, the magneto- hydrodynamic fluid motion of the wire plasma, the coupling of the wire plasma to the driving electrical circuit and the atomic physics and energetics of the plasma. The z-number, geometry and mass of the wire material also play an impor- tant role in the above process. A good deal of the under- standing of how these competing processes affect the dynamic and radiative behavior of the wire plasma is obtained from experimental data. In addition, computer codes such as WHYRAC [51] have been developed to calculate the detailed time history of the wire implosion. The codes provide in- Sight into the coupling of plasma to the generators and into the energetics of the implosion. Although many of the processes are fairly well understood, their relationship

and affect on one another is not.

It as theorized that one way to obtain x rays with’ the desired characteristics may be to use materials with differ- ent, higher z values or even combinations of materials.

Many materials can not be fabricated into thin wires or

do not have a high enough vapor pressure to allow the forma- tion of a vapor. In response, investigators have therefore proposed that the vapor puffs be generated by using a laser. One of the vacuum diodes would be fabricated out of the target material. The target electrode, irradiated by the

laser, would experience a steady temperature rise. If the


laser beam is intense enough, the surface of the target would melt and begin to vaporize. The vapor expands, exposing fresh target layers to heating and vaporization. Depending on the laser intensity, the vapor may remain un-ionized or a plasma may be generated. The high voltage, high current pulse from the Marx-generator would then be timed to pass through the cloud of plasma, neutral particles and molten metal. This procedure advances the possibility of using virtually any material or combination of materials, in addition to being able to vary the mass and temperature of the vapor cloud by varing the power and intensity of the laser pulse.

As indicated earlier, one important parameter on which the dynamics of the z-pinch and x-ray emission depends is the mass of the pinched column. This thesis investigates the mass of material which is blown off a surface as a func- tion of the laser parameters and a method by which it can be measured. Aluminum was selected as the initial target material because of the extensive work done with aluminum exploding wires, thus facilitating a comparison of the two

methods in the future.



The effect caused by the absorption of high power laser radiation on the surface of an opaque solid slab includes surface heating, melting and vaporization. If the laser intensity is low enough the vapor is un-ionized and remains transparent to the incoming laser radiation. At higher intensities, the vapor becomes ionized generating a plasma. Throughout this thesis, I will use the term "vapor" to de- note an ejected cloud of material which contains both neu- tral particles and plasma. In conjunction with the radiation impinging on the opaque surface, a cycle of events occurs. This is depicted in Figure 1. Prior to considering these events individually, a few comments on lasers in general are appropriate.

First, lasers can be divided into two categories, pulsed and continuous. Only pulsed lasers were considered in this study. Pulsed lasers, in turn, may be subdivided into normal pulse and Q-switched depending on pulSe operation.

A normal pulse laser is one which is pumped by a flashlamp and the radiation allowed to emerge when the threshold con- ditions for laSing are reached. Normal pulse lasers emit in pulses lasting tens of milliseconds with peak powers of He to 10> watts. The term Q-switched laser denotes lasers employing an element of variable loss within the cavity and

emitting peak powers greater than 108 watts and lasting


LASER BEAM —_—— a. |) i on a |



( paprarion Vwq SURFACE V\. boss y .- HEATING




Figure 1. Events diagram for laser-target interaction. Dotted boxes are considered negligible or do not contribute to the study at hand and will not be considered. For simplicity, feedback mechanisms between processes are not included.


several tens of nanoseconds. With Q-switched lasers, flux densities in excess of no watts/cm* are easily attained.

In this regime, the vaporization temperature of any metal will be reached in less than one nanosecond. At this point, the input energy begins to supply the latent heat of vapori- zation to a thin layer of material at the surface and break- down (avalanche lonization) will occur. During breakdown, the few naturally occurring thermal electrons from the target Surface are heated by the laser beam. They will gain energy until they are capable of ionizing neutral particles by collision. Each collision will produce ions and more elec- trons which will also be heated by the laser beam to the point where they too can ionize neutral particles. The process thus continues exponentially producing electrons and ions

and absorbing more and more of the laser radiation. The "breakdown threshold" is the laser intensity required to generate a high density plasma at the target surface. Experi- ments have shown that breakdown over a laser-irradiated sur- face exhibits a sharp threshold in laser intensity. Several models have been developed that predict this breakdown thres- hold [40,54]. Below threshold, insufficient energy is avail- able to propogate the avalanche ionization and the vapor remains relatively transparent. When the breakdown threshold 1s reached, a high absorption plasma cloud is formed on the target surface. For example, consider Q-switched neodymium-

glass laser radiation (1.06 um) incident upon an aluminum


target. <A typical pulse could have a flux of oe to 107°

watts/em- and a duration of 10-100 nanoseconds. Figure 2, taken from Ref. 4 models the relationship between the thres- hold intensity required for breakdown and the laser pulse duration. The prediction is that breakdown will occur in this case. Because of breakdown, the energy in the beam is devoted to heating a small amount of vaporized material to

a high temperature, while the heat transfer from the hot vapor to the bulk of the target material is limited due to the short pulse duration. A normal pulse laser has a series of spikes, many of which are capable of producing breakdown. The difference is that the individual spikes do not possess enough intensity to perpetuate the ionization process. The vapor is heated, breakdown occurs and absoprtion of energy increases, but the laser spike intensity falls off rapidly and breakdown ceases. This process is repeated many times resulting in more energy being coupled to the target and thus more material being ejected. Consequently, a given amount of energy delivered at very high power is less effective in causing vaporization than the same amount of energy delivered in a longer, lower power pulse. It is for this reason that

only normal pulse lasers were considered in this thesis.

A. REFLECTIVITY When one considers the coupling of laser radiation to the surface of a target, one first needs to know how much

energy is absorbed into the material. The Drude-Lorentz free




O nN



Oo th

Figure 2.

1o7-¢ oO"! O


Pulse duration time vs breakdown threshold intensity for an aluminum target irradiated by a 1.06 um laser pulse.

Alternatively, this graph gives the breakdown time vs peak value of incident laser beam intensity. Taken from Ref. 4.


electron theory provides an acceptable model for metals interacting with infrared wavelength photons [66]. According to the model, electromagnetic radiation interacts only with free electrons ina metal. The absorbed photon energy raises electrons to higher energy states in the conduction band. The excited electrons, in turn, collide with lattice phonons and other electrons as they give up their energy. The electron- phonon collision frequency 1S proportional to the phonon popu- lation in the metal. The phonon population, on the other hand, determines the temperature of the metal. Thus, the rise in temperature strongly affects the electron-phonon collision frequency which in turn affects the reflectivity of the metal. Based on the Drude-Lorentz theory several authors have developed theoretical expressions that predict the absorbtivity of metals [49,66]. |

The reflectivity of a "real metal" surface is largely an empirical matter. Bonch-Bruevich et al. [10] investigated the reflectivity at 1.06 um of aluminum, copper, dural, steel, and silver as the metals were irradiated with a laser beam. The investigators surrounded the sample with a sphere to monitor the reflected radiation, Figure 3. Pronounced changes both during each individual spike and over the pulse as a whole were reported. Figure 4 shows the generalized reflec- tivity behavior of a metal during an individual spike. The region of rapid change of reflectivity a-b is due to initial heating of the metal. The segment b-c has a constant reflectivity which the authors attributed to the constant temperature during the time

when the melting wave iS propagating into the metal. Zavecz [71],






Figure 3. Schematic of the apparatus used by Bonch-Bruevich, et al. [10] for measuring the change in reflecting power of a metal under the action of laser radiation.


\ \ 8-4 \ > \ -— \ ~ \ = ¢ } O 4 4 Te D LJ OF 2 0 2 4 6 8 | 2 TIME ps Figure 4. Change of reflecting power of silver at 1.06 um

during a single spike. The form of the laser spike with energy of 7.5 KJ/om? is also depicted [10].


disagreeing, feels that this low reflectivity plateau is

a function of the instantaneous pulse intensity and the Specimen's equilibrium temperature. Finally, the surface ee begins to rise again, thus causing a further drop in reflectivity, segment c-d. The increase in reflectivity to the right of point dis due to the decay of the spike

flux density and the subsequent drop in temperature. Similar results were published by Chun and Rose [21] as reflected in Figure 5. In addition, Chun and Rose examined the dependence Of absorptance on the depth (internal volume) of the laser crater. These results, depicted in Figure 6, indicate that absorptance changes as the depth of the crater changes. This means that even if the initial reflectivity is high, much of the energy in a focused normal pulse laser will be absorbed by a metal surface. Although only experimental results for incident radiation at 1.06 um has been described, experimental work at other wavelengths has been done [57]. In all cases the experimental observations show that under high-intensity

radiation, metal absorptivity increases non linearly.

B. HEATING OF THE MATERIAL Possibly one of the most important effects of intense laser irradiation is the conversion of optical energy in the beam into thermal energy in the material. The radiation mean free path for visible and infrared wavelengths in a solid 4

material such as a metal, is typically of the order of 10 ~ om

or less, so that the deposition of laser energy can be considered





Figure 5.


ae MO

6 8 |O

4 TIME- lOO ys

Time dependent reflectance measurement at 1.06 um from Ref. 21. A single laser pulse with a power density of 10/ W/cm* was used.


8 e O 9 oF é ; <a A = af G 4 a) =| | n 3 QO AL Or x NI © O cu Y) m+ ©O MO < J | 2 3 4 5 6 7 8 910


Figure 6. Dependence of absorbtance at iLO6ssimt On Cracer volume from Ref. 21. m/p is the ratio of mass removed to material density.


to be a surface phenomenon insofar as the transport of energy in a solid is concerned. The Drude-Lorentz free electron theory as previously discussed is the basis of thermal response of the material. The time required (relaxation time) for electrons, excited by irradiation, to transfer their energy to the lattice by means of electron-phonon collisions

a2o sec [36,72,74]. When compared

amounts to approximately 10 to a laser pulse, the relaxation time is short, allowing one to assume that the heat transfer to the solid is instantaneous because local equilibrium is rapidly established. Therefore, One is justified in assuming that temperature is a valid concept and so the normal equations of heat flow may be applied. The following development has been principally adapted from Harrach [30,32], but other authors have similar developments [7,49].

Suppose that at time t = 0 laser radiation is directed upon an Opaque, solid slab of metal. A portion of the radia- tion, iS absorbed; the rest being reflected as described in the previous section. A temperature distribution T(r,t) develops throughout the material. When a temperature gradient exists in a body, there is an energy transfer from the high- temperature region to the low-temperature region. The heat conduction equation (without phase changes) is [15]

caer ie)

Ved (r,t) + PC IE = A(r,t) (1)


where T(r,t) 1s the temperature, J(r,t) 1s the thermal energy crossing unit area per unit time, pc is the heat capacity per unit volume, and A(r,t) is the net heat energy per unit

volume time generated. Fourier's law relates the temperature

to the heat flux i @ ae 3) ee 8 OI ae we (2)

To simplify the calculations, it is assumed, as in Refs. 4 and 59, that the problem is planar, heat 1s introduced only from the surface and flows into the material in the x-direction. The thermophysical constants are independent of temperature,

radiation and convection from the surface is negligible, and

the liquid phase of the material can be ignored. The heat

flow can now be modeled using the Fourier heat-conduction

equation [54]

2 9“T(x,t) , 1 a


ee ce}

where k is the thermal diffusion constant equal to pKC and K is the thermal conductivity constant. The rate of heat

production can be written as

A = oCé (4)

where @ is the surface absorptivity, I is the flux density

of the incident radiation, 6 is the penetration skin depth of


radiation into the target material, op is the density of the target material, and C is the specific heat. Now by substi- tution of (4) into (3), the model equation becomes

oti e) at (x,t)

+ See SKE (5)

_ eeeedse) 2 at PCO


In order to solve this differential equation, the various parameters must be known or at least estimated. With the exception of §, each parameter has an accepted value which can be used. The value for § 1s not so easy to determine. It 1s expected to be small, 107° cm [74]. It can be assumed that the laser radiation is absorbed in an infinitesimally thin surface layer > 0). The source term may now be deleted from (5). The term is not lost however, since it

will be incorporated into the surface boundary conditions.

Equation (5) now becomes

3°T (x, t) 2


ats, c) Oe

- K + 0 (6) The specific boundary conditions depend on whether the sur-

face of the target 1s being vaporized or not. For the case

at hand, no vaporization, the surface boundary condition is

GPx se . Ete)

re = aC for x = 0




Ort) (7)

A second boundary condition arises when the target is con-

Sidered to be a semi-infinite solid.

Pim, obit X00 ox

And finally, the third condition is

Dexe0) = 0 foe,

In general, equation (6) does not yield an exact solution. One method of solution, utilized by Harrach, is the “heat balance integral method" [30]. This method, which is based On the assumption that "the partial differential equation is required to be satisfied only in an average sense throughout the solid rather than at each point x," is outlined below. The integral method reduces the nonlinear boundary value problem to an ordinary initial value problem where solutions can be expressed in closed analytical form.

Integrating eqution (6) over the spatial interval of the

target, Ko SxS & gives


= ax (10)

In general taking the derivative of an integral yields


g d = oT zs oe o f Tax = f ae ax + T(x=2,t) Fo Xs Xs Cid} ax _

Using equation (10) and the results of equation (11), noting

ad2/dt = 0 and assuming k is constant ss ox dT fi d S KD =| ) = => f T dx + T(x_,t) = iz) ox =) Sar ac 2 S ae

If the target surface is stationary then dx _/dt = 0. Goodman [26] points out that the key step in the careful choice of the solution form, T(x,t). The idea is to choose T(x,t) such that it can be integrated explicitly. This gives an ordinary First order differential equation when substituted into equation (12). Now to apply this technique to the problem defined by equation (6) and the boundary conditions (7),

(8), and (9)

kalI (t) (13)


Consider the solution form


T(0,t) [L-x/&(t)]* exp[-x/e(t)], (14)

BOT “0 <<. Ett)


EAM pC) ee Or aha ne

where T(0,t) is the front surface temperature and £&(t) 1s a

time dependent thermal penetration depth. Integrating

equation (14) over x from 0 to 2 and incorporating equation

(13) yields the ordinary differential equation SeIT(0,t)e(t)] = [kal (t)]/[1-2e7*]K (15) Substituting equation (14) into equation (7) gives E(t) = BRUCE) ale) (56)

Solving equations (15) and (16) simultaneously yields

t TOE ee =) (ee ey 2 (ere) (Tie )aet) 3(1-e2°")K 0




alte jac’ al({t)


Several special cases can now be considered by stipu- lating the laser profile I(t) and the dependence of the absorptivity on temperature, a = a(t).


Let the absorptivity coefficient be constant, aoe and the laser pulse be a step function. The solution to equation

(17) is now found to be

T(O,t) = ——-( ae uve (19) 3(1-2e 7) and from equation (18) E(t) = (3KT/(1-2e71)1/4 (20) For comparison, the exact solution as outlined by Carslaw and Jaeger [61] is 2al T(x,t) = 2 (RE) 1/2¢ 4, exp (-x?/4kt) 4(rkt) (21)

ms serfclx/(4nt)*/*]}

In the above equation erfc denots the complementary function


erfc(A) = Jj - erc(,)

A 2 = 2/n f exp(-u~) du 0

The agreement between the approximation, equation (19) and the exact solution, equation (21) seems to be quite good. For T(0,t), equation (21) is less than 1% higher. Now the solution just described applies up to a time t At T(O,t.) = T the vaporization time is 2

t= (3(1- 267) /kK}{KT /a lI} (22)

for = /3kr. > 1. O v=


The laser pulse is a step function and the absorption coefficient increases linearly with the surface temperature up to the vaporization temperature ue according to the

relation a = [a+ (a - ao) ](T(0,t)/TL]

The solution of the heat flow equation (14) at x = Q 1S

then given by

T(O0,t) = (a-a_)T /fa_-q (23)


with EA) = 3KT (a-d.)/lala -a)to] (24)

and t. is as before.


Material removal begins with vaporization. As vaporiza- tion proceeds, the absorption of the laser beam increases rapidly as the depth of the crater increases. At first the evaporation rate and boundary temperature are equal to zero and all of the laser radiation is used in heating the opaque solid as pointed out in the previous section. If the external flux 1s intense enough, the surface will be brought to the vaporization temperature ue at time te For later times, = > toe the problem is still defined by equation (6); but one of heat transfer in the presence of a moving phase boun- dary requiring new boundary conditions.

For the first boundary condition, the surface boundary energy balance relation, equation (7) must be changed to include a term to account for the energy expended on melting and vaporization. It must also account for the fact that the surface is no longer stationary.

—~k oT (x,t) = eG) = <1 —— (25)

OX = x=x_ (t)


where x. (t) 1s the instantaneous position of the surface. For example, x, = 0 whent<t,. L is the effective latent

heat of sublimation described by L = L The

+ : melt Lsapor second boundary condition 1s as before, that 1s the target

is assumed to be a semi-infinite solid

lim ——— = 0 (26)

The final condition is based on the temperature distribution

in the material at to.

Deane) -= Lis (27)

where the function f(x) is determined by the prevaporization solution. With the boundary conditions just described, equation

(6) becomes

: k i Wide) = “Seler(t) = ob

x S

If dx_/dt = Q then equation (28) reduces to the prevaporiza- tion form, equation (13), as expected. The trial solution

for the vaporization case may be chosen to be

x-x _ (t) ea)

ash) eee ~ Gives (ey! “SES ter



for 0 < x- x(t) < g(t) - x(t) and T(x,t) = 0 for x— x. (t) > E(t) ~ x(t). The surface temperature 1S fixed at T as the position of the surface x. (t) moves into the

solid. The surface recession velocity can be represented

by a(T )I(T_) CT : = Vv Vio yt, _ ae _ Vv Bee x, (t) = or 2 ie es a ie 5 (3 + 7 ey ee,

(30) When the material is exposed to large constant flux and begins vaporizing after time toe the rate of material removed will approach a steady state as the corresponding surface

recession velocity reaches a steady state.

Xs = a(T J I(T) /o (Lb + CT.) ene The steady state penetration depth, D(t) can now be written


Des = Ko L/KT = L/CT Figure 7 from Ref. 30 depicts the normalized thermal penetra- tion depth D(t), surface position x(t), and surface recession velocity x, (t) for aluminum. Next the crater depth de at

a specific time t. can be determined from the integral






Figure 7.

4 6 8 10 l2 I4 I6 NORMALIZED TIME (t/t )-1 Time dependence of normalized values of surface recession velocity x(t), thermal penetration

depth D(t), and surface position x _(t), for aluminum from Ref. 30. 2


20 = f x(t) dt (32)

assuming that the target is effectively infinitely thick.

The previous discussion indicates how the temperature of a local spot on a target can be raised to the point where vaporization begins. At the same time, thermal diffusion begins as the local spot temperature rises. As a result, for most metals, the material is removed as a consequence of three processeS: vaporization, the development of pressure in the cavity by vapor expansion and the creation of molten metal. Generally, vaporization begins at the surface. The initial velocity of a vapor jet flowing into a vacuum is equal to the local sound velocity. As the crater depth increases, the vapor velocity increases to a supersonic flow [4], which provides a mechanism for the washing of liquid metal from the walls of the crater. The fraction of material removed in the liquid state increases with pulse duration [21] because of the increase in temperature of the crater's inner walls with time.

Theoretical results are generally difficult to compare to experimental ones because of several factors. First, the amount of molten metal washed out of the crater and the initial fraction of absorbed energy is<