J . Phys. Chem. 1989, 93, 7509-7516 the probable distance between donors and acceptors indicates that the excitation-transfer rate will likely be very sensitive to the relative orientation of donor to acceptor. The transfer rate can be more accurately calculated by going beyond the dipole approximation to the monopole method4*once more structural information is known. The excitation-transfer rate from a J aggregate to a monomer acceptor molecule has been reported to be 100 times that of the monomer as a donor in monolayer assemblies of organic dye molecules.45 This mechanism would appear to be especially advantageous for the large antenna systems used by green bacteria, organisms that inhabit environments where only very weak light is available. (48) (a) London, F. J . Chem. Phys. 1942, 46, 305. (b) Chang, J. C. J. Chem. Phys. 1977, 67, 3901.
7509
Acknowledgment. This is publication number 12 from the Arizona State University Center for the Study of Early Events in Photosynthesis. The center is funded by U.S. Department of Energy Grant No. DE-FG02-88ER13969 as part of the USDA/DOE/NSF Plant Science Center program. This work was also supported by a Grant-in-Aid for scientific research from the Ministry of Education, Science and Culture, Japan, to M.M. (62540520), from the Biological Energy Storage program of the U.S. Department of Energy to R.E.B. (Grant DE-FG02-85ER13388), a grant from the U S . Department of Agriculture to R.S.K. (87-CRCR-1-2424), and the sponsors of the Laser Fusion Feasibility Project at the University of Rochester. Our thanks to Robert Hwang-Schweitzer for assisting on the streak camera measurements and George King for designing Figure 1A. Registry No. BChlc, 53986-51-9; BChla, 17499-98-8.
Theoretical Study of Ablative Photodecomposition in Polymeric Solids Bruce J. Palmer,* Thomas Keyes, Richard H. Clarke, and Jeffrey M. Isner Department of Chemistry, Boston University, Boston, Massachusetts 0221 5 (Received: July 13, 1988; In Final Form: April 12, 1989)
We present a detailed microscopic model of ablative photodecomposition in polymeric solids and show that it can be expressed in terms of a set of coupled rate equations. The principal feature of our model is the photochemical cleavage of bonds due to a nonlinear absorption event. Numerical simulation of the model shows that there is an intensity threshold below which no ablation takes place. Above the threshold ablation increases rapidly with laser intensity.
1. Introduction In the past several years, ablative photodecomposition of organic materials via high-intensity laser irradiation has been extensively investigated both as a potential therapy for patients with a variety of cardiovascular disorders and as a processing technique for a number of plastic substrates. Laser energy employed as a focused beam or transmitted via optical fibers has been demonstrated in vitro, in vivo in animal models, and more recently in human patients to be capable of ablating atherosclerotic plaq~e.I-~Work on plastics has shown that high-intensity irradiation can, under appropriate conditions, cleanly etch plastic substrates with little or no sign of thermal damage in the surrounding material.4d With regard to wavelength, previous studies have established that virtually any wavelength from the ultraviolet to the infrared regions may be used successfully to accomplish ablation of plaque. The principal determinant, however, of thermal damage resulting from laser ablation is the energy profile employed. Most biomedical applications of laser irradiation have utilized continuous wave (CW) energy delivery with typical outputs of well under 100 W. In contrast, pulsed lasers generate peak powers in the megawatt to kilowatt range with pulses of a few nanoseconds duration for most Nd:YAG and excimer lasers and a few microseconds for infrared and tunable dye lasers. At suitably high intensities, pulsed lasers can be used to eliminate thermal destruction at the ablation site in both biological tissue and plastics. (1) h e r , J. M.; Clarke, R. H. IEEE J . Quantum Electron. 1984, QE-20, 1406. (2) h e r , J . M.; Steg, P. G.; Clarke, R. H. IEEE J . Quantum Electron. 1987, QE-23, 1756. (3) Abela, G. S.;Seegar, J. M.; Barbieri, E.; Franzini, D.; Fenech, A,; Pepine, C. J.; Conti, C. R. J . A m . Coll. Cardiol. 1986, 8, 184. (4) Deutsch, T. F.; Geis, M. W. J . Appl. Phys. 1983, 54, 7201. (5) Braren, B.; Srinivasan, R. J. Yac. Sci. Technol., B 1985, 3, 913. (6) Koren, G.; Yeh, J. T. C. Appl. Phys. Lett. 1984, 44, 1112.
0022-3654/89/2093-7509$01 S O / O
At low intensities, even with pulsed lasers, a threshold is crossed and thermal damage is observed. Studies on polyimide and poly(methy1 methacrylate) films in the UV region have quantified the existence of this threshold. Plots of etch depth per pulse vs fluence for fmed pulse width consistently show a sharp increase in the etch depth per pulse at the intensity at which thermal damage disappears and clean ablation begins.”1° At even higher intensities there is another crossover that has been explained as screening of the incident beam by ejected material from the ablating solid, although a recent femtosecond study has cast some doubt on this interpretation.” However, the sharp increase in etch rate is a key feature in the characterization of ablation and one that we will be focusing on in the subsequent discussion. Further experiments have been aimed in elucidating the underlying mechanism by which ablation occurs. At present there are two major contenders for the primary mechanism, one thermal and one photochemical. The thermal mechanism postulates that the driving force is intense thermal heating of the substrate by the laser. Ablation is then the result of rapid thermal degradation of the sample. Because of the high intensity and short duration of the pulse, the region of temperature increase is confined to a sharply defined region near the surface of the sample and hence there is no thermal damage to the remaining sample. In the photochemical mechanism, ablation occurs because of bond cleavage by the absorption of one or more photons. If a suitable number of bonds are broken. the solid becomes unstable and (7) Srinivasan, R. Science 1986, 234, 559. (8) Srinivasan, R.; Braren, B.; Dreyfus, R. W. J. Appl. Phys. 1987, 61, 372. (9) Cole, H. S.;Liu,Y. S.;Philipp, H. R. Appl. Phys. Lett. 1986.48, 76. (10) Srinivasan, R.; Braren, B.; Seeger, D. E.; Dreyfus, R. W. Macromolecules 1986, 19, 916. (11) Kiiper, S.; Stuke, M. Appl. Phys. B 1987, 44, 199.
0 1989 American Chemical Society
7510 The Journal of Physical Chemistry, Vol. 93, No. 21, 1989
I
Palmer et al.
I
Figure 1. Schematic diagram of solid. Open circles represent monomer units, solid lines are covalent bonds between monomer units, and dotted lines are interchain van der Waals interactions.
molecular fragments begin to fly off the surface. Recent work on plastic substrates suggests that ablation cannot be uniformly accounted for as a purely thermal phenomena. Photoacoustic and calorimetric studies on polyimide films indicate that the rise in surface temperature of irradiated samples is on the order of 1000 K and that ablation occurs within 4-6 ns of the start of the laser p ~ l s e . I * - This ~ ~ is too short a time to be compatible with the known thermal degradation rate at this temperature. Time of flight measurements on the material ejected from the surface of irradiated samples also indicate a nonthermal contribution to the velocity distribution at high intensity.I5 At low intensities the only feature in the velocity distribution is a thermal peak corresponding to a temperature in the region of 1200 K. Above the ablation threshold intensity the velocity distribution broadens so that it is no longer Maxwellian and a secondary peak corresponding to fast particles appears and grows as a function of laser intensity. In spite of its many potential applications there has been relatively little theoretical work on photoablation. Jellinek and Srinivasan16and Keyes et al.” have presented simple rate equation models built around the idea of one-photon cleavage of chemical bonds. Sutcliffe and Srinivasan18 have expanded on this to incorporate the effects of a threshold intensity on the bond cleavage rate and Garrison and Srinivasan19have done molecular dynamics simulations comparing thermal and photochemical pictures of photoablation. However, no clear theoretical picture of photoablation has yet emerged. In this paper we present a rate equation approach to developing a microscopic description of photoablation. This allows us to begin formulating a detailed theory of photoablation and to compare the effects of different microscopic processes on the ablation threshold. 2. Photoablation Rate Equations In order to describe photoablation theoretically, it is first necessary to construct a model of the ablating solid. Since we are interested in the photoablation of organic materials such as plastics and tissue, we choose our solid to consist of long polymeric chains that interact with each other via weak intermolecular forces. The chains are composed of monomer units linked via covalent bonds. Because these are long chains, the number of bonds is equal to the number of monomers. If we had extensive covalent cross-linking between chains, the number of bonds would be greater than the number of monomers. Likewise, if the chains (12) Dyer, P. E.; Sidhu, J. J . Appl. Phys. 1985, 57, 1420. (13) Dyer, P. E.: Srinivasan, R. Appl. Phys. Lett. 1986, 48, 445. (14) Gorodetsky, G.; Kazyaka, T. G.; Melcher, R. L.; Srinivasan, R. Appl. Phys. Lett. 1985, 46, 828. ( 1 5 ) Danielzik, B.; Fabricus, N.; Rowekamp, M.; von der Linde, D. Appl. Phys. Lett. 1986, 48, 2 12. (16) Jellinek, H. H. G.; Srinivasan, R. J . Phys. Chem. 1984, 88, 3048. (17) Keyes, T.; Clarke, R. H.; h e r , J. M. J . Phys. Chem. 1985,89,4194. (18) Sutciiffe, E.; Srinivasan, R. J . Appl. Phys. 1986, 60, 3315. ( I 9) Garrison, 8.J.; Srinivasan, R. J . Appl. Phys. 1985, 57, 2909.
Figure 2. Schematic diagram of bond energy levels. Level I is bond ground state, level I1 is a vibrationally excited state of the bond, and a broken bond is represented by an excitation to the ground-state continuum. One- and two-photon absorption processes are represented by the arrows.
were short, the number of bonds would be less than the number of monomers since a relatively large number of monomers would be at the end of chains. The monomers themselves are assumed to have some internal structure so that the possibility of creating vibrationaly excited monomers exists. A schematic diagram of these chains, along with the weak interchain interactions, is shown in Figure 1. To model the absorption process, the covalent bonds are considered to be a two-level system, sketched in Figure 2. The lower level (level I) is the ground state of the covalent bond between monomers; the second level (level 11) is a vibrationally excited level and is separated from level I by the energy of one photon. Absorption of two photons puts the covalent bond into the ground-state continuum and effectively destroys the bond. Thus, bonds can be broken by two photon absorption from level I or one photon absorption from level 11. The excess energy from the photon goes into vibrationally exciting the monomers. This energy is initially fairly localized but eventually leaks into the normal modes of the monomer and from there into the remaining solid, at which point it is considered heat. The justification for a nonlinear absorption event comes from the time of flight measurements which point to some kind of crossover in the photochemistry at high intensity. This is reinforced by recent experiments by Natzie et al. on N O desorption?O where the absorption event is thought to be one photon. For this system the time of flight measurements reveal a nonthermal velocity distribution at low incident laser intensity with a small thermal component that increases with increasing laser intensity. This is qualitatively the reverse of what is seen in photoablation. As covalent bonds are broken the monomers near the surface will eventually become completely unbonded. They can then physically separate from the surface and fly off into space. We consider two detachment scenarios. In the first, the monomer remains attached to the surface via weak intermolecular interactions. It becomes detached upon absorption of an additional photon, with most of the energy going into translational energy of the outgoing monomer. This would account for the nonthermal distribution of ablated monomers. The second possibility is that the monomer simply flies off the surface when the last covalent bond attaching it to the remainder of the solid is broken. If the potential surface for the outgoing monomer is steep, this would again account for the nonthermal distribution of monomers. To describe this system mathematically, we set up a system of coupled rate equations for the densities of each of the quantities involved. For example, the number of ground-state bonds can be described as a space- and time-dependent density n,(x,t). We assume that at t = 0 the solid is located in the half-space x > 0 with the interface parallel to the yz plane. The only variation in the system is then on the x axis. Similarly, the number of excited bonds can be written as nII(x,t). In the undamaged solid the density n, has an initial value no. If we assume that the number of bonds in level I1 due to thermal excitation is negligible, then (20) Natzie, W. C.;Padowitz, D.; Sibener, S. J. J . Chem. Phys. 1988,88,
797s.
The Journal of Physical Chemistry, Vol. 93, No. 21, I989 751 1
Ablative Photodecomposition in Polymeric Solids the initial value for nII is zero. The rate equation for nI can be found by considering all processes that create and destroy level I bonds. These are as follows: (i) A ground-state bond (level I) can absord a single photon and become a vibrationally excited bond (level 11). (ii) A ground-state bond can simultaneously absorb two photons and be promoted to the ground-state continuum. This breaks the bond and results in the formation of excited monomers. (iii) Vibrationally excited bonds can relax to ground-state bonds via internal conversion. From this, the rate equation for nI is
anI/& = -k12nIZ- klcnIP+ kzlnII
(1)
The quantity Z = Z(x,t) is the local intensity and is proportional to the density of photons at the point x at time t . Equation 1 is entirely analogous to an ordinary chemical rate equation. k I 2 , k,,, and kzl are the corresponding rate coefficients for processes i, ii, and iii and the sign of the terms on the right-hand side of (1) is based on whether or not the process creates or destroys level I bonds. Similarly, we can write down a rate equation for level I1 bonds The first two terms in (2) correspond to the loss of level I1 bonds via internal conversion and absorption of a single photon into the ground-state continuum. The third term is a source term for level I1 bonds due to single-photon absorption by the ground-state (level I) bonds. Rate equations for the monomer density can also be written down. In this paper we distinguish between ground-state monomers, n,(x,t), and monomers that have been excited by a bond breaking event, nm*(x,t). We start with the equation for n,*. Excited monomers are created when a bond is broken and are destroyed when an excited monomer relaxes back into its ground state or is detached from the surface. If we ignore monomer detachment, then the rate equation for n,* is
an,*/at = klcnlP + k2cnIIZ- KmWmnm. The first two terms assume that we create one excited monomer per bond destroyed. This is a little unphysical since the deposition of energy from a bond breaking event is always asymetrically deposited on the monomers. We could eliminate this approximation, if necessary, by considering "doubly" excited monomers, nm**. The third term is due to internal conversion. The more difficult problem is to figure out the form of the terms corresponding to monomer detachment. In the undamaged solid we require two cuts in the polymer chain to separate a monomer from the rest of the polymer. The probability of a cut is 1 minus the actual number of bonds divided by the maximum possible number of bonds. The actual number of bonds is just NB = nI + nII. The maximum possible number of bonds is more complicated. In the undamaged solid the number of monomers and bonds is equal for polymer chains. The maximum possible number of bonds is then equal to the total number of monomers, NM = n, n,. However, as the number of monomers begins to decrease, the maximum possible number of bonds becomes less than the number of monomers since at low densities many monomer pairs will be too far apart to form bonds. The exact relationship between the maximum possible number of bonds and the number of monomers is a difficult problem, so we make the approximation that the maximum number of bonds is equal to the numbers of monomers for all monomer densities. If we define pB= NB/NM, then the probability that there is a cut at some point is (1 -pB). In the undamaged solid it takes two cuts to separate a monomer from the chain. As the number of monomers decreases, the number of cuts will also decrease since an increasing fraction of monomers will be found at the ends of chains. However, we again make an approximation and assume that two cuts are required to separate a monomer at all monomer densities. The monomer detachment is then proportional to
+
Since we are overestimating both the number of bonds per monomer and the number of cuts required to detach a monomer, this expression is smaller than the true detachment rate, except for the nearly undamaged solid. Another factor involved in monomer detachment is how close the monomer is to the surface. Monomers buried deep inside the solid will not become detached even if both bonds connecting it to the chain are cut. We expect this since the probability that a monomer will not collide with other monomers on the way out decreases exponentially as the average number of monomers between the unconnected monomer and the surface increases. This requires a modification of the detachment term to ( l - pB)zs(NM)nm*
where
A is a parameter that determines the distance inside the surface from which a monomer can escape. It is analogous to an inverse screening length. For the one photon detachment mechanism the rate equation for n,. is an,,/at = klcnIP + k2,nI1Z- ~ , . , n , ~ - K~,( 1 - pB)2s(NM)Zn,* (3)
For detachment when the last bond is cut the equation is an,*/& = kl,nIp + kzcnIIZ- Km*,nm* - Kej( 1 - pB)2s(NM)n,, (4) The only difference between eq 3 and 4 is the factor of Z in the last term of eq 3, which is not present in eq 4. The equations for n, are straightforward. We assume that the detachment mechanisms are the same as for n,. in particular, we assume that the coefficient K~~ is the same for both n, and n,*. We then arrive at the equations
Equation 5 is for one-photon detachment and eq 6 is for simple bond-breaking detachment. The first two terms in (5) and (6) reflect the fact that when an excited monomer is created by bond breaking, a ground-state monomer is removed from the system. So far we have ignored the possibility that bonds can re-form once they are broken. However, we would expect that bonds broken deep inside the solid should re-form fairly quickly since the monomers cannot go anywhere and remain properly aligned for recombination. In order to include this behavior we need to modify the rate equations for nI and nII. The probability that a bond re-forms is proportional to the probability that there is a cut times the probability that two fragments lie next to each other. If we assume that the bonds are re-formed with equal probability in levels I and I1 and that only monomers in their ground state re-form bonds, then the equations for nI and nII become
Bond re-formation does not affect the number of monomers, so eq 5 and 6 remain unchanged. Because of our assumption that every monomer can participate in two bonds at all monomer densities, we are overestimating the probability of there being a cut between two monomers and the recombination rate is too high. This will combine with our underestimation of the detachment rate to push the ablation threshold to higher intensity. To solve for the densities, we still need to find the intensity as a function of position. Following the derivation of Beers' law, we can calculate the change in intensity as a function of position
7512 The Journal of Physical Chemistry. Vol. 93, No. 21, 1989
by considering all processes that absorb photons. For the onephoton detachment mechanism this leads to the relation
ailax = -[tI2n1 + tlcInI + t2cn11+ tej(1 - ~ B ) ~ N M(9) ]I
Palmer et al. where Ax = xi - xi-l. We use the convention that any term whose argument is x - ~is defined to be zero. From this it follows that I’(xi) is given by
Note that eq 9 is in the form of an absorption equation with a nonlinear absorption coefficient” ar/ax = I(xJ can then be written as
This can be inverted to get I(x,t) =
I(xi) = Io(t)r-l(xi) exp[-~?(x,)]
exP[-xx dX’{[t12 + I t l c l n 1 + € 2 ~ ~ + 1 1 €ej(l -pB)’NM)] (10) Io(t) is the intensity as a function of time in the absence of any solid. The t’s are the extinction coefficients for the absorption processes already described. t12corresponds to one-photon absorption by a ground-state bond, t l c corresponds to two-photon absorption by a ground-state bond, t2ccorresponds to one-photon absorption by a level I1 bond, and tej corresponds to absorption by a surface monomer in the one-photon detachment mechanism. In the simple bond breaking picture the last term in (10) is absent and tej = 0. Equation 10 is not easy to work with directly because of the factor of I in the exponent. We therefore define the quantity r(x,t) by I r(x,t) = e x p ~0X e l c I ndx’ By taking the derivative of r(x,t) with respect to x and substituting for I(x,t) using eq 10, it follows that F(x,t) satisfies the first-order equation
ar/ax
=
clcnlIO(f) exP[-xx dx’le12nl +
tZcnII
+ € e j ( l -pB)2NM)]
(11)
We can easily integrate (1 1) to find r(x,t) and use the result in eq 10 to calculate I(x,t). These equations can be solved by using a straightforward algorithm. Since the solid is assumed to initially lie in the half space x 2 0, we need only worry about the positive x axis. We start by dividing the x axis from x = 0 to some value x = xmaxinto a set of M 1 evenly spaced points, xi, where xo = 0 and xM = x,,,. Each rate equation yields a set of M 1 equations representing the densities at the pointx xi. For example, the equation for nl becomes
+
+
anI(xi)/at = -klznl(xi)I(xi) - klcnl(xi)[I(xi)Iz + k21n11(xi)+ krec(1 - pB(xi))[nm(xi)12/2 (12) i = 0, 1, 2, ... M The remaining rate equations are all similar. We evaluate the time dependence by integrating the various densities as a function of time at each point xi using a fourth-order Runga-Kutta algorit hm. The only complicated part of this analysis is evaluating the intensity at each of the points xi. If we integrate both sides of eq 1 1 with respect to x we get r(x) - 1 =
xx
dx’tlcnIIo(t) exp[-Q(x?]
(13)
where we have defined Q(x) as ~ ( x =)
Jx
dx’ itizni
+w
I I +
tcj(l - M N M I
From the definition of r we get r ( x = 0) = 1 and hence the factor of 1 in (13). Using the trapezoid rule for the integral, we can write Q(xi) as Ax Q(xi) = j=o C-{[cI2nI(xj) + t2cnII(xj) + e e j ( l - pB(xj))2NM(xj)l 2 + [tlzn~(xj-l)+ tzcnII(xj-l) + tej(1 - p a ( x j - ~ ) ) ~ N ~ ( x j - ~(14) )II
(16)
3. Results and Discussion Because the model has a large number of parameters, it is imperative that we try and fix as many of them by physical considerations as possible. To systematically run through all possible values of the parameters is impractical since it would involve thousands of calculations. The first simplification we can make is to relate the rate coefficients for one- and two-photon absorption to the corresponding extinction coefficients. We sketch out the connection between klZand t l Z ,the remaining relations are all similar. The intensity, I, is a flux with units photons.cm-2.s-1 but we could also write it as a density times the speed of light, C
I =
Cnphoton
From Gauss’s theorem we know that the time rate of change of a density is equal to minus the divergence of the flux, which gives anphoton = --a i -
at ax The loss of photons is equal to the loss of level I bonds due to one-photon absorption and aI/ax = -qznII so klznlI = tlznII It follows that k12 =
€12
provided that I is in units of photons.cm-2.s-1 and the densities are number densities. Similarly k2c = t2c Kej = tej kIC = 2tlC The factor of 2 in the last relation reflects the fact that two photons are absorbed per bond broken. The relation between K~~ and tej only holds for the one photon detachment mechanism. For simple bond breaking detachment, tej = 0.0. We chose the density of monomers in the undamaged solid to be n, = 1.0 X loz2cm-). This corresponds to a molecular weight . of 50-100 g/mol for a solid with a density of 1 g ~ c m - ~The density of level I bonds in the undamaged solid is also given by no. The intensity of the laser beam Io(?) is switched from zero at t = 0 to some constant value Io for t > 0. This is equivalent to assuming a square wave pulse profile, while actual pulse profiles are more like a Gaussian. However, the results for a square pulse are easier to interpret and can easily be generalized to a Gaussian pulse. For the most plausible behavior obtained in this paper, the interfaces settle down to a linear motion very quickly. The instantaneous motion of an interface for a Gaussian pulse would closely follow the motion of the interface for a square pulse at the corresponding intensity. We look at values of Io in the neighborhood of photons-cm-2-s-1. Single-photon absorption in levels I and I1 is assumed to occur with equal probability so that tI2 = tk We set both of these equal to 1.0 X cm2. The decay length associated with this value is (t12n,,-l = 0.1 mm. The extinction coefficient for single-photon detachment is a little more complicated. In order for the interface to actually move, the detachment rate must be fairly high and this implies a large value for the extinction coefficient. A higher value for tej is plausible since it represents absorption by a species
-
Ablative Photodecomposition in Polymeric Solids
The Journal of Physical Chemistry, Vol. 93, No. 21, 1989 7513
with two dangling bonds, making it highly radical in character. We set tej= 1.0 X lo-'' cm2. If all the bonds were broken in the solid, without removing the monomers, then the corresponding decay length is 1000 A. This is comparable to absorption in a metal or plasma. The internal conversion rate coefficients kzl and K,., should have values comparable to vibrational relaxation times in a solid. These have been measured at 1-200 ps in a variety of ~ystems,2l-~~ so we use a value of 10 ps and set kZ1= xmtm = 1.0 X 10" s-l. Because steric factors in the solid should be highly favorable for bond recombination, the rate of bond recombination is assumed to occur on the same time scale as internal conversion, so that kr,nO k21. This leads to a value of k,, = 1.O X lo-" ~m~0s-l. In the one-photon detachment mechanism xej follows directly from tej,but for simple bond breaking detachment it must be determined separately. xej should correspond to a time on the order of a vibrational period, i.e. at the subpicosecond level. However, using a value in this region presents numerical difficulties since the time scale for detachment is then many orders of magnitude faster than the time scale for one-photon absorption. The rate equations become stiff in this regime and the time step used in numerically integrating them must be made small. In order to reduce the amount of time spent on computation, we have used slightly slower rates corresponding to lifetimes on the order of 10 ps and set xej = 1.0 x 10" s-l. The screening parameter X is chosen so that the decay length associated with monomer ejection is about lej 25 A. From the definition of S(NM) this implies Xlejno 1, so we set X = 4.0 X cm2. The only remaining coefficient is the two-photon absorption coefficient t l c which we treat as a purely adjustable parameter. For most calculations we set Ax = 1 A. The time increment was usually set at 10 ps, although for some of the high-intensity runs we used a value of 1 ps. Convergence was established by decreasing the space and time increments by a factor of 2 and rerunning the calculations. The results of the two calculations were then compared to see if there were any changes. No detectable changes where seen for nI and n,in any of the calculations shown here. A few of the remaining densities showed persistent fluctuations on the order of a few percent even after repeatedly decreasing the increment sizes. In particular, the quantities in Figure 4, except for n,, still showed slight variations on going to the smallest increments for which calculations remained practical. However, the variations were small and we estimate that the quantities shown are within a percent or so of the true values. In any case, the fluctuations have no effect on our conclusions. We start by examining the simple bond breaking detachment mechanism. In Figure 3 we show some two-dimensional plots of the level I bond density as a function of position and time. The density profile going into the solid is shown at successive time intervals after the laser is turned one. We look at a maximum distance of 1000 8, into the solid and for a period up to 10.0 ns. = 1.0 X cm2, tlc = 1.0 X For these calculations, t12= cm4-s,xej = kzl = K,*, = 1.O X 10' s-l, k,, = 1.O X lo-' c m 3 d , and X = 4.0 X cm2. The coefficient tlcwas chosen so that it gave a nice crossover from no ablation to ablation as the intensity Io passed through the value 1.0 X loz6 photonscm-2*s-1.The only difference between the calculations shown in Figure 3 is in the value of Io. In Figure 3a the intensity is set at a relatively low value of loz5photons-cm-2-s-1. At this intensity there is no visible change in the solid over the entire 10-ns duration of the calculation. In Figure 3b we increase the intensity by an photons-cm-2-s-1at which order of magnitude to Io = 1.0 X point we see an interface begin to form in nI. This is moving fairly slowly and has only etched out about 100 A after 10 ns, but otherwise it is sharp and well-defined. Furthermore, the speed at which the interface moves increases dramatically as we increase
1000 A
10 ns
A
10 ns
1000 A
-10 ns
1000 A
10 ns
-
1000
- -
(21) Heilwand, E. J.; Casassa, M. P.; Cavanaugh, R. R.; Stephenson, J. C. J . Chem. Phys. 1985,82, 5216. (22) Heilwand, E. J. Chem. Phys. Lett. 1986, 129, 48. (23) Maier, J. P.; Seilmeier, A.; Kaiser, W. Chem. Phys. Lett. 1980, 70, 591.
Figure 3. Density of level I bonds, nI,as a function of time and distance into the solid for the simple bond breaking detachment mechanism. The parameters are cI2 = c2c = 1.0 x cm2, clc = 1.0 x cm4-s, ~~j - kzl = K , , , . ~ = 1.0 X 10" s-l, k , = 1.0 X lo-" cm3&, and X = 4.0 X cm2. Note that all curves have the constant value nI = no at t = 0, which determines the vertical scale. (a) Io = 1.0 X lo2' photons. cm-2.s-1. (b) Io = 1.0 X photons-cm-2-s-1. (c) Io = 2.0 X photons.cm-2.s-1. (d) Io = 4.0 X photons.cm-2d.
the intensity. In Figure 3c the intensity is raised by a factor of 2 to 2.0 X loz6photons.cm-2-s-1. The interface moves considerably faster and about 500 have been etched away at the end of 10 ns. Increasing the intensity even further, to Io = 4.0 X photons.cm-2-s-1in Figure 3d results in an interface that has moved the full 1000 A distance shown in the figure after a few nanoseconds. These calculations clearly show that at low intensities there is no ablation of the solid and that once ablation starts it increases rapidly with laser intensity. In addition to the density nI we can look at the other densities in the system. For the values of the parameters shown in Figure 3c we plot the remaining densities nII,n,, and nm* in Figure 4. In addition we plot the spatially dependent monomer detachment ) N is ~ the . rate at which rate, Rdet = xej(l - P ~ ) ~ S ( N ~This monomers are being removed from the system and should be localized at the front of the interface. From Figure 4 it can be seen that nI and n, are virtually identical. This is a consequence of the fact that the bond re-formation rate is fairly high. The densities nIIand nm+ are several orders of magnitude smaller and quite different in appearance from nI and n,. These show a maximum in the vicinity of the interface and then decay sharply into the solid. The reason that nIIand nm+are so small compared to nI and nm+ is that the relaxation processes due to internal conversion are much faster than the time scales for one photon absorption and motion of the interface. This keeps the "steady state" concentrations of level I1 bonds and excited monomers low. We are also interested in the effect of the screening parameter X on ablation. In Figure 5 we repeat the calculations shown in Figure 3 except that we set X = 0.0. This means that monomers
7514
The Journal of Physical Chemistry, Vol. 93, No. 21, 1989
Palmer et al.
1000
A
10 ns
1000
A
10 ns
1000 A . 1000 li -
-10ns
10 ns
1000
-10ns
(4
“;“..bik:
0 A
A
I
11s
(4
Figure 4. Other densities corresponding to conditions in Figure 3c. (a) Density of vibrationally excited bonds, nII. Vertical scale has been increased by a factor of 50 relative to scale in Figure 3c. (b) Local monomer detachment rate, Rdet = ~ ~ - .~~ B( ) ~1 ~ ( N M )The N Munits . are arbitrary. However, compare this with Figure 6 . (c) Density of ground-state monomers, n,. The vertical scale is determined by n, = no at t = 0. (d) Density of excited monomers, nm*. Vertical scale has been increased by a factor of 20 relative to scale in Figure 3c.
with both bonds cut can escape no matter how deep inside the solid they are located. This may correspond, in a crude sense, to the mechanism of Garrison and S r i n i ~ a s a nwhich ~ ~ postulates that if enough bonds are cut the resulting internal strains cause the material to become mechanically unstable. Ablation results when the system tries to relieve the stress by ejecting material. However, we see in Figure 5 that there is relatively little change in the behavior of the system. The biggest difference is that the interface moves slightly faster relative to the corresponding conditions in Figure 3. If we look at the local monomer ejection rate, shown in Figure 6, for the interfaces in parts a and c of Figure 5, we see that monomer ejection appears to extend unphysically far into the solid in Figure 6a. However, the scale of Figure 6a is about 5 orders of magnitude smaller than Figure 6b so that even though the monomer ejection rate is unphysically distributed, it is so small that it does not make any difference. At higher intensities, monomer ejection is mostly localized at the interface, even without invoking the screening length A. For these parameters the value of X does not drastically alter the qualitative behavior of the system. The motion of the interface is, however, strongly influenced by the value of the two photon absorption coefficient, clc. This is not surprising since we have been assuming that ablation is driven by a nonlinear absorption event. Dropping the two-photon absorption coefficient 2 orders of magnitude to clc = 1.0 X cm4-s slows the motion of the interface considerably and also pushes up the threshold intensity at which ablation appears. In Figure 7 we show two interfaces for clc = 1.0 X cm4-s, Io
Figure 5. Density of level I bonds as a function of time and distance into the solid for the simple bond breaking detachment mechanism. All parameters are the same as for Figure 3 except the inverse screening length which has been set at X = 0.0. The vertical scale is determined by nl = no at t = 0. (a) Zo = 1.0 X photons.cm-2*s-i. (b) Io = 1.0 X photons-cm-2.s-1. (c) Io = 2.0 X photons.cm-2d. (d) Io = 4.0 X 1026photons.cm-2.s-1.
1000
A
-10 ns
Figure 6. Local monomer detachment rate, &et = Kej( 1 - PB)F(NM)NM, as a function of time and distance into the solid for the calculations shown in Figure 5. (a) Rdetfor conditions shown in Figure 5a. Vertical scale has been increased by a factor of lo5 relative to scale in part b. (b) &et for conditions shown in Figure 5c. Same scale as Figure 4b.
= 1.0 X
photonscm-2d and two values of A, X = 0.0 and cm2. All other parameters are the same as in Figure 3. If we drop the intensity by an order of magnitude to I, = 1.0 X photons.cm-2d, then no significant ablation occurs. In addition to the change in threshold intensity and interface speed, the parameter X plays a bigger role in the system. Dropping X to zero results in a very broad interface but also one cm2. that moves much faster than for X = 4.0 X X = 4.0 X
The Journal of Physical Chemistry, Vol. 93, No. 21, 1989 7515
Ablative Photodecomposition in Polymeric Solids
1000
A
-
A
1000
A -
1000 A
10 ns
'
-10 ns
10 ns
1000
1000
A
10 ns
1000
A
10 ns
.4
-10 ns
1000
~
-10 ns
Figure 7. Density of level I bonds, nl, as a function of time and distance into the solid for the simple bond breaking detachment mechanism. The cm2,tlc = 1.0 x IO* cm4.s, ~~j parameters are t I 2 = tZc= 1.0 x = kZ1= K,., = 1.0 x 1Ol1 s-l, k,, = 1.0 X lo-" c m 3 d , and Io = 1.0 X photons-cm-2-s-1.The vertical scale is determined by nl = no at cm2. (b) X = 0.0. t = 0. (a) X = 4.0 X
1000 A
1,000
Figure 8. Density of level I bonds, nI,and ground-state monomers, n,, for k , = 0.0 and different values of t l c (simple bond breaking detachment mechanism). Other parameters are t I 2 = tlc = 1.0 X cm2, K~~ = k21 = K,., = 1.0 x 10" s-', X = 4.0 X cm2, and Io = 1.0 X photons.cm-2d. The vertical scale is determined by nI = no at t = cm4-s. cm4*s. (b) n, for tIc = 1.0 X 0. (a) nl for t l c = 1.0 X cm4-s. cm4.s. (d) n, for tIc = 1.0 X (c) nI for t l c = 1.0 X
To determine what effect bond re-formation has on the system, we repeated some of the calculations with k , = 0.0. We started by fixing the intensity at 1.O X photons-cm-2-s-1 and increasing the two photon absorption coefficient, el,, in powers of 10. The value of X was set at X = 4.0 X cm2. In Figure 8 we show nI and n, for t l c = 1.O X cm4*sand el, = 1.O X loA2cm4*s (in Figure 9a,b we also show a calculation for elc = 1.0 X loA4 cm4*s). It is immediately obvious that the densities nI and n, do
A
-
10 ns
Figure 9. Density of level I bonds, nI, and ground-state monomers, n,, for k , = 0.0 and different values of the intensity (simple bond breaking detachment mechanism). Other parameters are t I 2= t2c= 1.0 X cm2,K~~ = k2, = K,., = 1.0 X 10" s-l, X = 4.0 X cm2, and t l c = 1.0 X loa4 cm4.s. The vertical scale is determined by nl = n, = no at 1 = 0. (a) nl for Io = 1.0 X photons.cm-2.s-1. (b) n, for Zo = 1.0 photons.cm-2.s-1. (d) X photons.cm-2*s-1.(c) nl for Io = 4.0 X photons.cm-2.s-1. n, for Io = 4.0 X
not coincide. For the lower value of tlc a relatively sharp interface forms in the density of monomers but not in the density of ground-state bonds. This means that the region near the interface is composed of completely unbonded monomers on a length scale of at least 100 8, or so. This is physically unreasonable since such a material should be unstable even on the short time scales involved here. The fact that nI and n, do not coincide is caused by the nonzero value of A. If we set X = 0.0, then nI is unchanged and n, resembles nI. This still creates a problem with the lower value of el,, since the interface is now too broad but the higher value of qcproduces a reasonable looking surface. We also tried fixing el, and varying the intensity. Shown in Figure 9 are calculations for a system with el, = 1.O X lo4 cm4-s, X = 1.0 X cm2, and the remaining parameters as in Figure 3. We did calculations at two intensities, Io = 1.0 X phophotons-cm-2-s-1. The lower tons*cm-2-s-1and Io = 4.0 X intensity calculation cal also be considered part of the series shown in Figure 8. When the, intensity is increased, bonds are destroyed uniformly across the 1000-8, region shown in the figure. The density of monomers, q,,,shows a fairly sharp interface but it again suffers from the problem that there is no correspondence between n, and nI. We can eliminate the difference between nI and n, by setting X = 0.0, but the interface is then very broad. The conclusion from all this is that bond re-formation plays an important role in producing a sharp interface. Finally, we examine the one-photon detachment mechanism. In Figure 10 we show some calculations for t12= t2, = 1.O X cm4*s,tej = 1.0 x cm2, k,,, = 1.0 X cm2, el, = 1.0 x lo-" cm3.s-l, and k21 = K,,,., = 1.0 X 10" s-l. Shown are in-
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The Journal of Physical Chemistry, Vol. 93, No. 21, 1989
Palmer et al.
2-o
1.5 1.o
0.0 1000
A -
10 ns
Figure 10. Density of level I bonds, nI, as a function of time and distance into the solid for the one-photon detachment mechanism. The parameters are t 1 2= tZc= 1.O x cm2, elc = 1.0 x cm4-s, tej = i.O-I7 cm2, kll = K,,,.,,, = 1.0 X 10" s-l, k , = 1.0 X lo-" c m 3 d , and Io = 1.0 X photons.cm-2.s-1. The vertical scale is determined by nI = n, at t = 0. (a) X = 4.0 X cm2. (b) X = 0.0.
terfaces for both X = 4.0 X cm2 and X = 0.0. The intensity is set at Io = 1.O X 1027photons.cm-2-s-'. If we drop the intensity an order of magnitude, there is almost no detectable ablation. Both values of X result in relatively sharp interfaces although the X = 0.0 interface moves considerably faster. The most notable changes in going from simple bond breaking detachment to one-photon detachment is that the threshold for ablation is pushed to slightly higher intensities and the motion of the interface is slowed down relative to comparable intensities for simple bond breaking detachment. Because of the uncertainty involved in many of the parameters in this model, it is difficult to draw solid conclusions. However, we do get a few indications about what may be important in photoablation. If the multiphoton picture of bond breaking is correct, then ablation appears to represent a competition between bond breaking by two-photon absorption and bond recombination. At low intensity, bond recombination is rapid enough to maintain a low concentration of broken bonds and monomers cannot become detached from the solid. As the intensity increases, the rate of bond breaking eventually overwelms the rate of bond recombination, the density of broken bonds jumps up, monomers begin to fly off the surface at a significant rate, and ablation occurs. The detachment mechanism is also important since it determines at what density of broken bonds significant monomer detachment takes place. Another interesting point is the lack of any induction period in our model. Once the intensity threshold is crossed, ablation sets in almost immediately. However, several researchers have noted that a number of induction pulses are needed before ablation occurs.11*18 A possible explanation has been that a couple of pulses are required in order to break up the solid enough for ablation to start. This explanation seems implausible if we include the effects of bond re-formation since bonds re-form rapidly compared to the time delay between pulses (up to 1 s). Without bond recombination it is hard to produce either a sensible solid or a sharp interface. In addition, it is difficult to rationalize why bond recombination can be neglected altogether. An alternative explanation is to assume that the first few pulses result in a chemical modification of the solid and that it is this modified solid in which ablation takes place. This has been proposed by Kiiper and Stuke,' who postulate the creation of incubation sites that could
'
1
I
1
0
P
I
P
A
O I
I
I
23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0
Figure 11. Etch depth as a function of log [I07] for 10-ns pulses (circles) and 100-ns pulses (triangles).
lead to enhanced absorption and thus help lower the ablation threshold after a few pulses. These effects could be included in a straightforward way within this formalism, but a clearer physical picture of what these incubation sites consist of would be desirable. Presumably, further experiments will be able to shed some light on this issue. Another area in which our model appears to do badly is in predicting the etching behavior as a function of constant fluence (constant energy per unit area per pulse). Recent experiments by Taylor et al.24show that there is very little change in the etch depth per pulse for the same fluence but different pulse width for 7 and 300 ns (there is, however, a large change in going from nanosecond to picosecond pulses). This indicates that the total energy deposited is more important than the maximum intensity in this regime. Unfortunately, our numerical algorithms are not yet good enough to explore this question fully. In order to make a complete comparison, we need to be able to do calculations for values of Io at least an order of magnitude above the threshold intensity, and this is at the very limit of our computational ability. We can do a crude comparison in the region of the ablation threshold. The etch depth per pulse is plotted as a function of laser fluence for the data shown in Figure 3. If the width for a square pulse is defined as T, then the fluence is proportional to I07. The etch depth per pulse was evaluated by calculating the interface motion near the end of the pulse so that the transients due to the sharp interface in the initial condition have died out. We then extrapolate this to a 10-ns pulse. We also include a point for Io = 1.O X 1027photons.cm-2-s-1 which is not shown in Figure 3. Since the interfaces in Figure 3 all settle down quickly to a linear motion, the etch depth for a 100-ns pulse of intensity Io can also be obtained from the data in Figure 3. A few points for 100-ns pulses are also included in Figure 11. It is clear that the results depend heavily on pulse width, while experimentally there is very little dependence for pulse widths in this range. The most likely explanation is that thermal processes are beginning to play a role. We are currently working on incorporating temperature into this formalism and plan on investigating these effects in a future paper. Acknowledgment. We are indebted to Drs. Tom Gauthier and Ed. Gaffney for many discussions and to the referees for several useful suggestions. (24) Taylor, R. S.;Singleton, D. L.; Paraskevopoulos, G . Appl. Phys. Lett. 1987, 50, 1779.
