Thermal and Fluid Processes of a Thin Melt Zone during Femtosecond Laser Ablation of Glass: The Formation of Rims by Single Laser Pulses.

Transcription

1 Preprint for submission to Journal of Applied Physics Thermal and Fluid Processes of a Thin Melt Zone during Femtosecond Laser Ablation of Glass: The Formation of Rims by Single Laser Pulses Adela Ben-Yakar, 1, Anthony Harkin, 2 Jacqueline Ashmore, 2 Robert L. Byer, 1 and Howard A. Stone 2 1 Applied Physics Department, Ginzton Lab, Stanford University CA, USA Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA (Dated: September 29, 2003) Abstract We study the formation mechanism of rims created around femtosecond laser-ablated craters on glass. Experimental studies of the surface morphology reveal that a thin rim is formed around the smooth craters and is raised above the undamaged surface by about nm. To investigate the mechanism of rim formation following a single ultrafast laser pulse, we perform a one-dimensional theoretical analysis of the thermal and fluid processes involved in the ablation process. The results indicate the existence of a very thin melted zone below the surface and suggest that the rim is formed by the high pressure plasma producing a pressure-driven fluid motion of the molten material outwards from the center of the crater. The heat transfer calculations show a melt thickness of the order of 1 µm and a melt lifetime of the order of 1 µs. The numerical solution of fluid motion of the thin melt, on the other hand, demonstrate that the time for the melt to flow to the crater edge and to form a rim is about 1 µs, which is of the same order of magnitude as the melt lifetime. The possibility of controlling or suppressing the rim formation is also discussed. adela@stanford.edu; tel: (650) , fax: (650)

2 I. INTRODUCTION Laser micromachining has been widely studied for many materials and has recently been extended to ultrashort laser pulses that can machine any material to very high precision at the micron scale [1]. The key benefits of an ultrashort pulse include its ability to produce a very high peak intensity (> W/cm 2 ) and rapid deposition of energy into the material. While high peak intensities allow energy delivery even into transparent high band-gap materials such as glass, the rapid absorption of energy allows material removal, or ablation, before significant heating of the substrate occurs. Ultrashort pulses interact with the material by a nonlinear mechanism that is very different from that of conventional longer pulse lasers. The high energy density during the ultrashort laser pulses can initiate nonlinear processes such as multiphoton absorption [2, 3], which allows machining of any material independent of the laser wavelength. For example, it is possible to micromachine glass with very high precision using near infrared wavelengths, around 800 nm, at which the linear absorption properties of glass are very weak. While the nonlinear effects and short time scales associated with ultrashort laser ablation are believed to provide a non-thermal micromachining of glass materials, in some cases complicated surface morphologies result [4, 5]. Detailed experimental studies have shown that the laser ablation process might actually be thermal in nature, even in the femtosecond operating range [5]. Our preliminary estimates show that a very small amount of energy, less than 3% of the incident energy, might be deposited in the undamaged part of the borosilicate glass as thermal energy and so create a transient shallow molten zone below the ablated area (where there is the expanding plasma) [5]. The molten material rearranges (flows) due to forces acting at the surface of the melt. Another recent study also measured that about 8% of the incoming energy was thermalized and transmitted to the undamaged part of a quartz material when irradiated with an ultrafast laser pulse [6]. In order to improve the quality and precision of ultrafast laser micromachining, it is important to understand, model and quantify the thermal nature of the process and any material rearrangement that occurs. In this paper, we provide scaling arguments for the depth of the melted layer and model its motion during the lifetime of the melt. If the melt lifetime is long enough, forces acting on the liquid will be able to drive the molten material from the center to the edges of the crater, which creates an elevated rim around the ablated 2

3 crater. Multiple laser pulses show that this rim formation causes a surface roughness and therefore reduces the precision of an ultrafast laser micromachining process. Two main forces might affect the flow of a molten layer below the expanding plasma: 1) thermocapillary forces (Marangoni flow) [7, 8] and 2) forces exerted by the pressure of the plasma above the surface [9, 10]. Thermocapillary flow is induced by the temperature gradient on the surface which is expected to follow the Gaussian beam intensity profile of the laser. In studies of laser texturing of silicon surfaces in the absence of ablation, the rim formation was attributed to the thermocapillary flow in thin films created by nanosecond laser pulse heating [7]. The temperature gradient on the surface creates surface tension gradients that drives material from the hot center to the cold periphery. This response is expected in most materials where the surface tension, γ, decreases as the fluid gets hotter (dγ/dt < 0). However, in the case of glass dγ/dt is positive [11]. Consequently, such a thermocapillary flow in laser irradiated glass surfaces would actually drive fluid from the cold periphery to the hot center of the melt in contrast to what was observed in our experiments. In addition, the effect of thermocapillary flow in glass is expected to be negligible because of its high viscosity, which leads to a flow time scale much longer than the typical melt time scale, as detailed further below. On the other hand, a hydrodynamic force due to the pressure gradients exerted by the plasma onto the molten material might play an important role in the rim formation. A gradient of ablation pressure on the molten surface can induce a lateral melt flow to the periphery. The pressure gradients are expected to be particularly large at the edges of the ablated crater, which should be close the plasma/air interface. Because of these large pressure gradients, we might expect a melt flow to the periphery and rise of a thin rim at the edges of the melted surface much like a splash of a fluid. Although some of aspects of this mechanism have been discussed qualitatively in the literature for nanosecond laser irradiation of metals [9], in this paper we provide a detailed model of the flow processes associated with the first pulse and so give the time evolution for the profile of the melt surface. The mathematical model of the physical processes gives rise to a rim topography, though has one significant unknown which is the pressure gradient along the surface that results from the plasma. This uncertainty primarily affects time scales and not the physics. We first present typical experimental results of the surface morphology of craters ablated using single and multiple femtosecond laser pulses with an emphasis on the rim formation 3

4 around the craters (Section II). We then discuss the level of the material heating and estimate the variation of melt depth with time for a single laser pulse (Section III A). Finally, we present a thin film model to estimate the characteristic time scales of various mechanisms causing the melt to flow outwards to create the rim and show representative numerical simulations of fluid flow in a thin molten layer (Section III B). We conclude with some ideas for suppressing formation of the rim (Section IV). II. EXPERIMENTAL RESULTS Our experimental studies focus on the surface morphology of borosilicate glass (Borofloat TM ) ablated using single femtosecond laser pulses. We irradiated the glass samples with 800-nm 100-fs pulses from a regeneratively amplified Ti:sapphire laser. The laser was focused with a 250-mm focal-length lens to a spot size of about 30 µm. The surface of the sample was positioned to be normal to the direction of the incident beam. We performed these experiments in air at atmospheric pressure. Following irradiation, the samples were analyzed with a scanning electron microscope (SEM). The basic results were reported recently [5] and we summarize them here with some additional features as they form the basis for the theoretical and modeling considerations given in Section III. Figure 1 presents three SEM images of crater rims produced by an average laser fluence of 34 J/cm 2. The first image (Fig. 1a) shows a thin circular rim around a nearly smooth crater following a single pulse of the laser on a flat glass substrate. It is the resemblance of this rim to a resolidified splash of a molten layer that originally motivated our investigation of the dynamical processes that result in the formation of the rim. The second image (Fig. 1b) shows that when a second pulse irradiates a previously formed rim, a new rim is formed inside the original rim. The distance between the two rims is approximately equal to the wavelength-λ of the light, which suggests that diffraction of light plays an important role [12]. Upon close inspection of the area between the pulses, there is a wave pattern apparently due to the diffraction-induced modulation of the second laser pulse. When a third pulse irradiates two previously overlapping craters (Fig. 1c), micrometerscale organized features appear along the rim. The smaller scale ripple-like features that are evident circumferentially along the rim, which is basically semicircular in cross-section, are 4

5 presumably a manifestation of a Rayleigh-capillary instability familiar from the disintegration of fluid filaments. This aspect of the surface evolution has not been studied. As shown in Fig. 2, the interplay between rim formation and diffraction results in a perceptible surface roughness when attempting to micromachine channels in a glass substrate by scanning a pulsed laser across the surface. A detailed AFM study of a single crater is shown in Fig. 3. The rim is raised by about nm above the surface and the maximum depth of the ablated crater is about nm depending on the laser fluence [13]. In order to understand and control the micromachining process using ultrafast lasers, it is necessary to investigate the formation mechanism of the surface microfeatures. To address these issues, we first have to examine the rim formation of a single laser pulse, which is the main focus of this paper. III. THEORETICAL MODELLING AND DISCUSSION A theoretical model is described to characterize the thermal and flow processes associated with the first laser pulse. The goal is to identify the nature of the rim formation around the laser ablated craters and suggest ways (such as modification of the laser profile) to eliminate it. The laser deposits energy to the substrate which melts the glass. To estimate the melt thickness, and its variation in time, the optical light penetration and heat conduction into the material are studied (Section III A). A thin film model of the molten surface layer is then introduced to explore the temporal evolution of the melt surface (Section III B). A. Heat Transfer Calculations In this section, we estimate the thickness and the lifetime of the molten layer and present a model for the time dependent variations of the melt thickness. To achieve this, we first calculate the fraction of the laser energy deposited in the bulk of the material as heat by analyzing 1) the fraction of the incoming energy that is absorbed by the material and 2) how this energy dissipates from the illuminated regime (i.e., partition sequence of the absorbed laser energy). In dielectric materials, the incident laser beam is absorbed by electrons through nonlinear 5

6 processes (multiphoton and avalanche ionization) [2, 3] and then transferred to the lattice on the time scale of few picoseconds. At this stage, a high pressure plasma is formed above the surface; the plasma expands and removes the ablated material away from the surface. Dissipation of the energy deposited in the substrate begins only after the laser pulse is gone and when the energy is transferred from the high energy electrons to the lattice. Numerical simulations of the fluid and thermal dynamics of a laser-induced plasma on an aluminum target irradiated with 100 picosecond laser pulses show that a few microseconds after irradiation, 70 % of the absorbed energy is used by the expanding plasma to move the ambient gas [14]. About 20 % of the absorbed energy is lost in radiation and only 10 % of it remains in the target as thermal energy. In the case of femtosecond laser ablation of glass, a rather complicated and time consuming numerical solution is required to estimate the partition of the absorbed laser energy. In this paper, we propose to use the optical light penetration depth to estimate the fraction of the incoming laser energy deposited in the glass as heat (thermal energy) [13]. 1. Initial Energy Deposition Depth We assume that the absorbed laser energy is deposited uniformly in a layer defined by the penetration of light as illustrated in Fig. 4. The light intensity is attenuated exponentially with depth z according to the Beer-Lambert law, which provides a convenient means to quantify the penetration depth of the absorbed laser energy. Then the absorbed laser fluence as a function of depth, F a (z), is given by where A is the surface absorptivity, α 1 eff F a (z) = AF 0 exp ( z ) α 1, (1) eff is the effective optical penetration depth, and F 0 is the average laser fluence (energy per unit area). Let us next summarize how we determine the values of A and α 1 eff from experimental measurements. The absorptivity of glass depends on the intensity of the laser irradiation and varies with time during the duration of the laser pulse. When glass is exposed to high intensity ultrashort pulses, its reflectivity rises with time as the plasma density increases [2]. Once the critical surface plasma density is formed, any further incident laser energy is reflected back from the surface due to an induced skin effect. Perry et al. [2] showed that when 6

7 the incident laser fluence is high above threshold (F > 5 10F th where F th 2 J/cm 2 for fused silica), a plasma with a critical density can be achieved early in the pulse and a large portion of the energy is reflected back from the sample. At a laser irradiance of W/cm 2 they calculate the reflectance of fused silica to be around 70%. We used a similar laser irradiance, and therefore we assume that on average 30% of the incident energy is absorbed by the borosilicate glass, namely A = 0.3. For the effective optical penetration depth, we use experimental data from our previous measurements [13]. We determined α 1 eff = 224 nm by calculating the slope of the linear relationship between the ablation depth, h a, and the logarithm of the incident laser fluence, F 0, expressed by ( ) h a = α 1 eff ln F0. (2) F th Here, F th = 1.7 J/cm 2 is the measured threshold fluence for borosilicate glass when 200-fs and 780-nm laser pulses are used. Since at laser fluences below F th no ablation occurs, the material is ablated to a depth, z = h a, where the laser fluence drops to F th. Therefore, by substituting F a (z = h a ) = AF th in Eq. (1) we can obtain the expression given in Eq. (2). The distance, where the laser fluence decreases to 1/e of its value, is interpreted as the effective optical penetration depth (α 1 eff ) in accordance with the Beer-Lambert absorption law. Using Eqs. (1) and (2), we can write the variation of the absorbed laser fluence with depth, F a (z), in terms of F th and h a, F a (z) = AF th exp ( z h ) a α 1, (3) eff where z = 0 corresponds to the location of the flat surface away from the irradiated zone. We assume that the absorbed laser energy that penetrates beyond the ablation depth, h a, remains in the material as thermal energy. Since the laser fluence at the ablation depth is equal to AF th, the thermalized energy remaining in the material as heat is E heat = AF th (πw 2 0), where w 0 is the 1/e 2 laser beam radius at the surface. Thus, we find that a fixed amount of laser fluence, approximately F heat = AF th = 0.5 J/cm 2, heats the undamaged part of the material independent of the incident laser fluence when it is well above the threshold. This means that, in our experiments, presented in Fig. 1, less than 2% of the total incident laser pulse energy goes into heating of the material (the value F heat = 0.5 J/cm 2 is about 1.5 % of the incident laser fluence of F heat = 34 J/cm 2 ). 7

8 2. Initial Temperature Distribution The initial energy deposition described by Eq. (3) produces an initial temperature distribution in the glass, T (z, 0), with an exponential profile T (z, 0) T = F a ρc p α 1 eff = F heat ρc p α 1 exp eff ( z h ) a α 1 eff (4) where F heat = AF th is the portion of the absorbed laser fluence that goes into heating, T is the initial temperature, ρ is the density of the substrate, and C p is the heat capacity. Melting occurs when the local temperature exceeds the temperature, T m 1500 K [15]. The value T m used here is the working point temperature of glass, defined as the temperature at which the glass can readily be formed, which corresponds to a viscosity of 10 3 Pa.s. Since the glass materials do not have a latent heat of melting, all of the absorbed energy goes into melting. We can therefore calculate the initial melting depth, h m (t = 0) = z m h a, from h m = α 1 eff ln(f heat/(t m T )ρc p α 1 eff ) where C p = 1250 J/kgK (see the thermophysical properties of glass in Table I); we have chosen to evaluate all physical parameters at the mean temperature of 900 K. This calculation yields an initial molten layer thickness of h m (t = 0) 424 nm. In the above discussion of energy deposition, we assume that the incident laser beam and the resulting initial temperature distribution inside the glass material attenuate with depth in accordance with the Beer-Lambert law (see the illustration in Fig. 4). When the material is illuminated with an ultrashort laser pulse, a high-temperature and high-pressure plasma is formed above the surface, presumably down to the ablation depth. The ablated material is removed by the expansion of the plasma. Below the plasma there is a thin zone of a molten glass with an initial thickness of h m = z(t m ) h a that can be calculated using Eq. (4). The molten region grows, then decays according to conduction process and the shape of the molten layer changes. Both these effects are described in more detail below. 3. Heat Conduction and Variation of Melt Thickness Following the cessation of ultrafast energy input, the melting process continues as the heat flows out of the region where the initial energy is deposited. Diffusion of the thermal energy determines the movement of the melting front and therefore the variation of the melt thickness. 8

9 We calculate the heat flow out of the region where the initial energy is deposited by solving a one-dimensional (1-D) heat conduction equation with an initial temperature profile described by Eq. (4). The cooling at the top of the melt zone is assumed to be negligible because of the presence of the high temperature plasma. During the expansion, the plasma cools in tens of microseconds from some very high initial temperature to the ambient temperature. So all of the heat loss is assumed to take place through the solid, as the rearrangement of the molten layer takes place on shorter time scales. The one-dimensional heat conduction model can then be described by T t = T (D z z ), z > h a (5) T z = 0 at z = h a (6) T = T as z (7) T (z, 0) = T + F heat ρc p α 1 eff e (z ha)/α 1 eff (8) where h a is the ablation depth or, in other words, the depth where the molten region begins. As a first approximation, we assume that the heat capacity, C p, the thermal conductivity, k, the density, ρ, and therefore the thermal diffusivity, D = k/ρc p, are constants at an average temperature of 900 K. To find an analytical solution for this heat conduction problem, nondimensionalize variables as: T = ρc pα 1 eff (T T ), z = z h a F heat The nondimensional problem statement is then α 1 eff, t = D (α 1 eff )2 t. (9) T t = 2 T, z 2 z > 0 (10) T = 0 z at z = 0 (11) T = 0 as z (12) T ( z, 0) = f( z) = e z, (13) which has the solution T ( z, t) = 1 4π t 0 [ ] f(ζ) e ( z+ζ)2 /4 t + e ( z ζ)2 /4 t dζ. (14) 9

10 The solution for f(ζ) = e ζ can be expressed in terms of the complementary error function, erfc(s), as: [ ( ) ( )] T ( z, t) e t z + 2 t z + 2 t = e z erfc + e z erfc. (15) 2 4 t 4 t We can now calculate the temperature distribution in the material that is not ablated (i.e. below the plasma) using the analytical solution given in Eq. (15). The variation of the melt depth as a function of time follows from h m (t) = z m h a = α 1 eff z as where T (z m, t) = T m. A plot of the melt depth as a function of time is shown in Fig. 5 for two different laser fluences thermalized in the material as heat; 1) F heat = AF th = 0.5 J/cm 2 and 2) F heat = 1.0 J/cm 2 since it is possible that radiation from the high temperature plasma also contributes to the melting of the material. The numerical results show that for the material parameters representative of the experiment, the melting process continues for µs and then solidification begins. These calculations provide us estimates for the order of magnitude of two important characteristic scales of the melt zone: An average melt depth, h m, of the order of 1 µm. An average melt lifetime, t m, of the order of 1 µs. With these estimates in hand, we will discuss in Section B which hydrodynamic forces acting on the melt have enough time to rearrange the liquid to form a rim within the melt lifetime. B. Fluid Dynamics Calculations 1. Thin Film Model In order to examine hydrodynamic conditions under which a crater partially filled with molten glass can form rims reaching the heights measured in experiments, we develop a one-dimensional model of the fluid motion as illustrated in Fig. 6. It is natural to work with cylindrical coordinates. The free surface of the molten glass is described by z = h(r, t) and the boundary between the liquid and the solid substrate is given by the time-independent profile z = b(r). I don t know quite how to state the justification for this. - Tony The shape of the thin liquid film evolves owing to variations in the surface tension γ along the free surface and because of the very high pressure p pl (r) of the plasma above the free surface. Because the melted region has a typical height much less than the typical width of 10

11 the laser ablated craters, the lubrication approximation (see Appendix A) can be used to obtain an evolution equation for the time-dependent profile of the free surface h(r, t): h t + 1 ( ( r r r (h b)2 dγ (h b)3 dp pl (h b)3 γ + r h )) = 0, (16) 2µ dr 3µ dr 3µ r r r r } {{ } } {{ } } {{ } Marangoni Pressure Curvature where µ is the liquid viscosity and the different flux contributions have been labelled. The first term in brackets accounts for motion caused by the surface tension gradients due to the uneven heating of the surface (thermocapillary or Marangoni-driven flow). The second term accounts for the motion due to the pressure gradients exerted by the plasma onto the molten material, and the third term in (16) represents surface tension effects, which enter as the product of surface tension and surface curvature (the latter has been linearized). Also, because of the very complicated dynamics of the plasma, for which the details are unknown, we assume in our model that the plasma pressure field, p pl, at the surface of the molten material is independent of time and therefore we use a time-averaged plasma pressure distribution, i.e. p pl (r) only. There are three assumptions needed to justify the use of Eq. (16): (i) The film must be thin, i.e. h m /L 1, where h m is the average melt depth and L is typical of the radial dimensions of the molten region. (ii) The Reynolds number R e for the film flow (see Appendix A) must be small, R e = ρ h m 3 p pl /(µ 2 L) 1. (iii) The flow must be quasisteady which requires the time for viscous diffusion across the thin layer, ρ h m 2 /µ, to be small compared to the time scale for film evolution. We verify these assumptions below. When a surface temperature distribution, T s (r), is imposed, then the surface tension varies according to We assume γ T dγ dr = dγ dt s dt s dr. (17) = dγ/dt s is constant and neglect the effect of temperature variations on the viscosity. Using Eq. (16) we can estimate characteristic time scales associated with the Marangoni flow (τ M ), and the pressure-driven flow (τ p ), Marangoni flow: Pressure-driven flow: µl 2 τ M γ T T m h m (18) µl 2 τ p p pl h m 2 (19) 11

12 where h m is an average melt depth, L is a typical radial dimension, and p pl is an average plasma pressure. Table I. The thermophysical properties of borosilicate glass are summarized in For an average melt depth of h m 1 µm, as estimated in the heat transfer calculation in Section A, a typical crater radius of L = 5µm, and an average plasma pressure of p pl 1000 atm (the typical plasma pressure drops from millions of atmospheres to about 10 atm during the first microsecond of its expansion [16]), we obtain τ p 10 4 sec and τ M 10 1 sec. It should be noted that using L = 5µm gives timescales for τ p and τ M that are too slow by one or two orders of magnitude in cases where the plasma pressure gradient, p pl /L, is very large only near the edge of the crater. In such cases, the characteristic length scale L would better be described by the much shorter distance over which the pressure gradient changes. In any event, we have ( ) τ M ppl h m = O 10 3 (20) τ p γ T T m and hence the characteristic time scale for Marangoni flow is two to three orders of magnitude longer than that of pressure-driven flow, and τ p τ M even if the peak pressure is lowered more than a factor of ten. It is clear from this estimate that the large plasma pressure above the free surface acts to move the fluid much more quickly than do surface tension gradients. This result suggests that once an expanding plasma is formed above the glass surface during laser ablation [? ], the spatially varying plasma pressure will control the evolution of the free surface at the interface. This mechanism for rim formation contrasts with that in laser texturing of silicon surfaces in the absence of ablation, which is attributed to the Marangoni flow in thin films created by nanosecond laser pulse heating [7]. The idea that thermocapillary effects do not contribute to the observed formation of a rim at the edge of the melt zone in our experiments is supported by the observation that the surface tension coefficient of borosilicate glass is positive [11] in contrast to the usual negative values of most pure liquids. Therefore, as discussed earlier, thermocapillary (Marangoni) flow in laser irradiated glass surfaces would be expected to drive fluid from the cold periphery to the hot center of the melt, which is not what is observed in the experiments shown in Fig. 1. We close by verifying the three assumptions used to obtain Eq. (16). First, the molten region must be thin h m /L 1. Since h m 1 µm and L 5µm, this condition is reasonably well satisfied. In fact, errors in using Eq. (16) are O (( h m /L) 2 ), which further 12

13 justifies the use of the lubrication approximation for this thin film flow. Second, we require that the effective Reynolds number, R e, for the film flow must be small. Using typical parameter values and viscosity at T = 2000 K we obtain ( ) ρ hm 3 p pl R e = O (21) µ 2 L Note that even if the pressure gradient that drives the flow was increased three orders of magnitude, the low-reynolds-number assumption, equation (21), is still satisfied. Finally, the third assumption requires that the viscous effects must act quickly to establish the velocity profile across the thin region. This time is O(ρ h m 2 /µ) s, which is much faster than the time scale, τ p, of fluid motion. Hence, the three principle assumptions in the fluid dynamics calculation are verified. 2. Numerical Results and Discussion for Pressure-Driven Flow To perform numerical simulations of the evolution of the molten glass we first nondimensionalize the time, length, height, and plasma pressure by characteristic values S = t/τ p, R = r/l, H = h/ h m, B = b/ h m, P pl = p pl / p pl. Since the key estimate of the previous section, Eq. (20), indicates plasma pressure gradients primarily drive the fluid flow, then we take γ = constant (the value is given in Table I), neglect the Marangoni term from Eq. (16), and arrive at the following nondimensional evolution equation for the free surface height, H(R, S), H S 1 [ (H B) 3 R R R dp pl B)3 + Λ(H 3 dr 3 The nondimensional parameter Λ is given by R Λ = γ h m p pl L ( 1 R ( R H ))] = 0. (22) R R Even though Λ is small, we will keep the curvature term in the numerical simulations. We also assume for simplicity that the melt-solid boundary is time independent, i.e., B(R) only. In Fig. 7 and Fig. 8 we numerically explore the behavior of solutions of Eq. (22). The rim height is the feature of the solutions in which we are most interested. More specifically, we investigate how changing the plasma pressure profile, P pl (R), or modifying the substrate 13

14 shape, B(R), affect the rim height. Since the plasma pressure and bottom substrate are not well characterized experimentally, then it is important to at least understand how changes in these quantities affect the crater and rim formation. Figure 7 shows the numerical solutions of pressure-driven melt flow for two different plasma pressure profiles. It is not well understood how the plasma pressure distribution depends upon the laser beam profile, and because of this uncertainty, we studied the effect of two different pressure profiles, one with a Gaussian distribution and another with a steeper gradient at the edges. The solid curve in the top plot describes a pressure profile of P pl (R) = exp( R 10 ) and the dashed curve is the gentler distribution P pl (R) = exp( R 2 ). For the free surface and bottom substrate profiles used in the numerical simulations, we require that the initial thickness of molten glass in the center of the crater reflects a reasonable value from the melt depth plot of Fig. 5. Therefore, the free surface shape of the molten glass in the numerical simulations is initially described as H(R, 0) = 0.5 exp( 15R 10 ), and the bottom substrate is given by B(R) = exp( R 10 ). At the center of the crater, R = 0, the melt depth is H(0, 0) B(0) 0.4, or roughly 0.4µm, which based on Fig. 5 is a reasonable melt depth. Solving Eq. (22) numerically for each of the plasma pressure profiles to a dimensionless time of S = 0.08 gives the two solutions shown in the bottom plot of Fig. 7 (solid and dashed curves). The rim height of the solid curve solution is larger than that of the dashed curve solution. These rim heights are consistent with the plasma pressure gradient of the solid curve in the top plot being larger than that of the dashed pressure curve. Converting the rim heights to dimensional values gives heights of roughly 210nm for the solid rim and 65nm for the dashed rim after a time of roughly 2µs. There is consistency between this range of rim heights, the time-scale of their formations, and rim heights observed experimentally, though it should be noted that the plasma pressure is not known with a large degree of certainty. We believe that the order of magnitude is reasonable and so the qualitative and quantitative trends offer insight into the details of the film rearrangement. In Fig. 8 we explore how changing the shape of the bottom substrate, B(R), while keeping all other parameters the same, affects the rim height. A plasma pressure profile of P pl (R) = exp( R 10 ) was chosen for the two simulations to be performed. The initial shape of the free surface is again given by H(R, 0) = 0.5 exp( 15R 10 ). For the first simulation (dashed curve), the bottom shape was chosen to be B(R) = exp( R 10 ), which 14

15 reproduces the solid curve solution in Fig. 7. The bottom shape for the second simulation (solid curve) is given by B(R) = exp( 0.3R 20 ), which provides more fluid at the edges of the domain than in the first simulation. The solutions are shown at S = 0.04, or roughly 1µs. The solution corresponding to the solid curve has a taller rim height than that of the dashed curve. The reason for this is that the fluid flux is proportional to (H B) 3, as seen in Eq. (22), and there was more fluid available to be transported in the second simulation. So, although we do not know exactly how the melt thickness changes in the experiments, these simulations show that a rim of a reasonable height may still be formed within the order of magnitude of the melt lifetime and that the typical results are independent of the precise forms of the plasma and bottom profiles. As a final remark, we note that the formation of transient rim-like structures occur in other fluid dynamics problems, such as the splash of drop on a solid surface or on a liquid layer (e.g. the famous Edgerton photograph of a splashing drop of milk). In particular, in the case of drop splash, numerical calculations for thin-film dynamics in the absence of viscous effects, which is very different from the dynamical situation we discuss, are reported by, among others, Yarin and Weiss [17], and experiments with drops splashing following their impact with thin layers are reported in Shin and McMahon [18]. IV. HOW TO SUPPRESS THE RIM FORMATION? We can now answer the question of how to suppress the rim formation for clean laser processing or in other words seek how to achieve clean borders of the irradiated spot. If the pressure-driven flow time, τ p, is long enough, the pressure induced forces acting on the fluid will not be able to drive the molten material from the center to the edges of the crater during the melt lifetime. From the parametric dependence shown in Eq. (19), τ p depends on the viscosity, µ, the average thickness of the molten layer, h m, and the average pressure above the surface, p pl. Smaller melt thickness or smaller average pressures above the surface lead to longer times for melt flow. The melt lifetime t m (α 1 eff )2 /D and therefore a thinner melt thickness can be achieved by reducing the optical penetration depth, α 1 eff (e.g. using shorter laser wavelengths or/and shorter pulse durations). Both the melt thickness and lifetime will be further reduced by the decreased amount of the absorbed thermal energy (F heat F th ) because the threshold fluence decreases with both laser wavelength and pulse 15

16 duration [19, 20]. In addition, materials with higher thermal diffusivities can reduce the melt lifetime by more rapidly removing the absorbed thermal energy. Another way of suppressing the rim formation may be modifying the plasma pressure profile. As shown in Fig. 7, with a less steep Gaussian pressure profile the formation of a rim takes longer time. It may be possible that for even a less steep pressure gradient near the edge, a rim cannot be formed during the lifetime of the molten material. Therefore, if one can modify the pressure profile, by modifying for example the laser spatial beam profile, it may be possible to achieve a cleaner border of the irradiated spot. Although it is easy to control the intensity distribution of the laser beam profile, it is not clear, however, how this may impact the pressure distribution. This remains a subject for further investigation. V. CONCLUSIONS The morphology of the single-shot ablated areas revealed a smooth and shallow crater surrounded by an elevated rim. From these experimental observations, conclusions could be drawn about the ablation mechanism of borosilicate glass. We argued that a very thin melt zone existed during the ablation process and calculated the thermal and flow properties of this thin melt zone. In these calculations, several characteristic time constants associated with ablation, melting, and flow processes were determined. The comparative values revealed that a flow of fluid driven by a plasma-induced pressure gradient localized near the radius of the laser pulse, with reasonable values of the plasma pressure, would have enough time to move melted material towards the edge and so deposit a thin rim around the ablated area. Physically based estimates of the melt lifetime and the pressure-driven flow process then suggest ways to suppress this rim formation. Acknowledgments The work has been supported by the TRW research fund and the Harvard NSEC. The authors gratefully acknowledge the contributions of Prof. Eric Mazur, Dr. Mengyan Shen, and Dr. Catherine Crouch to this investigation. 16

17 VI. APPENDIX APPENDIX A: DERIVATION OF THE EQUATION OF MOTION FOR THE THIN FILM MODEL In this appendix we present a short derivation of the partial differential equation describing viscous fluid flow in a thin film [21 23]. Suppose that the position of the free surface of the fluid is denoted z = h(x, y, t), and the shape of the time-independent bottom substrate is denoted z = b(x, y). We assume that the flow is incompressible and described by the Navier-Stokes equations. Then, the fluid velocity u and pressure p satisfy u t + u u = 1 ρ p + µ ρ 2 u, u = 0. (A1) (A2) For some of the estimates below, it is convenient to denote the mean film thickness h m = O(h b) and the average, or typical, fluid pressure by p. Note that in (A1) we are neglecting the gravitational body force since for the small length scales characteristic of the rims in the experiments, ρgh m / p 1. The boundary conditions to be satisfied are: no slip on the solid substrate, the normal and tangential stress balances across the free surface, and the kinematic boundary condition on the free surface. For pressure-driven flows on the scale L typical of the flow direction, we expect a typical velocity along the film to have magnitude u = O(h 2 m p /(µl), in which case we define the Reynolds number for the thin-film flow as R e = ρh 3 m p /(µ 2 L). Let us suppose first that the velocity field is u(x, y, z, t) = (u, v, w). Then, under the thin-film (lubrication) approximation we assume h m /L 1 and inertial effects are negligible, which is equivalent to the requirement that the Reynolds number is small, R e 1, so that the Navier-Stokes and continuity equations reduce to 2 p = µ 2 u 2 z 2 p z = 0 2 u 2 + w z = 0 (A3) (A4) (A5) where 2 = ( x, y ) and u 2 = (u, v). Within the lubrication approximation, the boundary conditions are 17

18 where p pl p = p pl γκ on z = h (normal stress) (A6) µ u z = 2γ on z = h (tangential stress) (A7) u = v = w = 0 on z = b (no slip) (A8) h t + u 2 2 h w = 0 on z = h (kinematic condition) (A9) is the plasma pressure above the free surface, γ is the surface tension of the interface, and κ is twice the mean curvature of the interface. We have assumed p pl to be time-independent in the numerical simulations reported in this paper. Equation (A4) and the normal stress balance give the local pressure in the liquid to be p = p pl γκ, b(x, y) z h(x, y, t) (A10) and we will use a linearized expression for the mean curvature term, κ = 2 2h. Substituting the pressure into Eq. (A3), integrating the result twice with respect to z, and using the no-slip and tangential stress boundary conditions yields u 2 = 1 ( ) z 2 µ 2 b2 ( 2 + bh zh p pl 2 (γ h)) + 1 µ (z b) 2γ. (A11) Integrating the continuity equation shows that h t + 2 q = 0 with q = z=h z=b u 2 dz, (A12) where q is the flux vector. Substituting (A11) for u 2 to calculate the depth-averaged flux, q, we then obtain the evolution equation for the height, h, of the thin film: [ h (h b) 2 t + (h b)3 ( 2 2 γ 2 p pl 2 (γ 2 2 2µ 3µ h))] = 0. (A13) In cylindrical coordinates and assuming an axisymmetric shape, h(r, t), Eq. (A13) becomes h t + 1 [ ( ( (h b) 2 r r r dγ (h b)3 dp pl (h b)3 γ + r h ))] = 0. (A14) 2µ dr 3µ dr 3µ r r r r 18

19 [1] J. Meijer, K. Du, A. Gillner, D. Hoffmann, V. S. Kovalenko, T. Masuzawa, and A. Ostendorf, Annals of the CIRP 51/2/2002, 1 (2002). [2] M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, J. of Appl. Phys. 85, 6803 (1999). [3] C. B. Schaffer, A. Brodeur, and E. Mazur, Meas. Sci. Technol. 12, 1784 (2001). [4] S. Ameer-Beg, W. Perrie, S. Rathbone, W. Wright, J. Weaver, and H. Champoux, Appl. Surface Phys , 875 (1998). [5] A. Ben-Yakar, R. L. Byer, A. Harkin, J. Ashmore, H. A. Stone, M. Shen, and E. Mazur, submitted to Appl. Phys. Lett. (2003). [6] F. Ladieu, P. Martin, and S. Guizard, Appl. Phys. Lett. 81, 957 (2002). [7] T. Schwarz-Selinger, D. G. Cahill, S. C. Chen, S. J. Moon, and C. P. Grigoropoulos, Phys. Rev. B 64, (1999). [8] D. A. Willis and X. Xu, J. Heat Transfer 122, 763 (2000). [9] V. N. Tokarev and A. F. H. Kaplan, J. Phys. D: Appl. Phys. 32, 1526 (1999). [10] A. Ben-Yakar, A. Harkin, J. Ashmore, M. Shen, E. Mazur, R. L. Byer, and H. A. Stone, Proceedings of the SPIE, The International Society for Optical Engineering 4977 (2003). [11] W. D. Kingery, J. Am. Ceram. Soc. 42, 6 (1959). [12] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7th ed. [13] A. Ben-Yakar and R. L. Byer, Proceedings of the SPIE, The International Society for Optical Engineering 4637, 212 (2002), also submitted to J. of Appl. Phys. [14] F. Vidal, S. Laville, B. Le Drogoff, T. W. Johnston, M. Chaker, O. Barthélemy, J. Margot, and M. Sabsabi (Annual meeting of OSA - Optical Society of America, 2001). [15] R. H. Doremus, Glass Science (John Wiley and Sons Inc., New York, 1994), 2nd ed. [16] F. Vidal, S. Laville, T. W. Johnston, O. Barthelemy, M. Chaker, B. Le Drogoff, J. Margot, and M. Sabsabi, Spectrochimica Acta Part B 56, 973 (2001). [17] A. L. Yarin and D. A. Weiss, J. Fluid Mech. 283, 141 (1995). [18] J. Shin and T. A. McMahon, Phys. Fluids 2, 1312 (1990). [19] B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, Phys. Rev. B 53, 1746 (1996). 19

20 [20] T. Q. Jia, Z. Z. Xu, R. X. Li, B. Shai, and F. L. Zhao, Appl. Phys. Lett. 82, 4382 (2003). [21] A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys. 69, 931 (1997). [22] T. G. Myers, SIAM Rev. 40, 441 (1998). [23] H. A. Stone, Nonlinear PDE s in Condensed Matter and Reactive Flows (2002), p [24] R. W. Cahn, P. Haasen, and E. J. Kramer, Materials Science and Technology: A Comprehensive Treatment (John Wiley and Sons Inc., 1998). [25] R. L. David, ed., CRC Handbook of Chemistry and Physics (CRC Press, ), 80th ed. 20

21 APPENDIX B: LIST OF TABLES Table I Thermophysical properties of borosilicate glass (Borofloat TM ). The chemical composition includes 81% SiO 2, 13% B 2 O 3, 2% Al 2 O 3, 4% Na 2 O. 21

22 APPENDIX C: LIST OF FIGURES FIG. 1 SEM images of crater rims generated by (a) one laser pulse (b) two overlapping laser pulses, and (c) three overlapping laser pulses of 800-nm and 100-fs. The laser fluence was F 0 = 34 J/cm 2. The numbers correspond to the order of the incident laser pulses. FIG. 2 An SEM image of a microchannel created using 200-fs and 780-nm laser pulses of F 0 = 23 J/cm 2 focused to a spot size of about 12 µm. The image shows the microscale surface roughness created by the rims of overlapping laser pulses when scanning the laser across the surface. FIG. 3 An AFM study of a single crater ablated with a 200-fs and 780-nm laser pulse of F 0 = 12.6 J/cm 2. (a) The AFM image. b) The ablation profile at the center-line of the crater. The ablation depth, h a, is normalized by an average melt depth of h m = 1 µm and the radial distance, r, is normalized by a characteristic radial dimension of L = 5 µm. FIG. 4 Absorbed laser energy deposition according to the Beer-Lambert law. A sketch of the resulting initial temperature distribution inside the material is shown along with definitions of several variables used in the analysis. FIG. 5 Melt depth variation with time for two different laser fluences using the approximation that % of the incident laser energy remains in the glass as thermal energy. The calculations are performed assuming constant thermophysical properties at an average temperature of 900 K. FIG. 6 Description of parameters used in the thin film model to calculate the evolution of the surface of the melt. FIG. 7 Numerical solutions of pressure-driven melt flow, described by Eq. (22), for two different pressure profiles. The plasma pressures are shown in the top plot where the solid curve is given by P pl (R) = exp( R 10 ), and the dashed curve is P pl (R) = exp( R 2 ). The bottom plot shows the evolution of the free surface for each of the pressure profiles. FIG. 8 Numerical solutions of pressure-driven melt flow for two different bottom substrate shapes. The bottom plot shows the evolution of the free surface for each of the 22

23 substrate profiles (solid and dashed curves). 23

24 TABLE I: Thermophysical properties of borosilicate glass (Borofloat TM ). The chemical composition includes 81% SiO 2, 13% B 2 O 3, 2% Al 2 O 3, 4% Na 2 O. Property Symbol Units Values Density [15] ρ kg/m Melting T m K 1500 temperature [15] a Viscosity [15] µ Pa s 10 3 (at 1500 K) 10 2 (at 2000 K) Surface tension [24] γ N/m 0.28 (at 300 K) Temperature coefficient γ T N/m K of surface tension [11] Thermal k W/m K 1.25 (at 300 K) conductivity [24] b 1.6 (at 600 K) 4.5 (at 900 K) 42.0 (at 1500 K) Specific heat [25] b C p J/kg K 746 (at 300 K) 1000 (at 600 K) 1250 (at 900 K) 1320 (at 1500 K) Thermal D m 2 /s (300 K) diffusivity b (600 K) (900 K) (1500 K) Ben-Yakar et al. a T m is the working point temperature of glass, defined as the temperature at which the glass can readily be formed and has a viscosity of approximately 10 3 Pa.s. b In the heat conduction calculations, we used data at an average temperature of 900 K, which is an average between T = 300 K and T m = 1500 K. 24

25 (a) x 3, m 2 (b) 1 x 4, m 4 3 (c) 3 pulses overlapping 2 2 pulses overlapping 1 x 4, m FIG. 1: SEM images of crater rims generated by (a) one laser pulse (b) two overlapping laser pulses, and (c) three overlapping laser pulses of 800-nm and 100-fs. The laser fluence was F 0 = 34 J/cm 2. The numbers correspond to the order of the incident laser pulses. Ben-Yakar et al. 25

26 x 5,000 5 m FIG. 2: An SEM image of a microchannel created using 200-fs and 780-nm laser pulses of F 0 = 23 J/cm 2 focused to a spot size of about 12 µm. The image shows the microscale surface roughness created by the rims of overlapping laser pulses when scanning the laser across the surface. Ben-Yakar et al. 26

27 (a) 0.2 (b) 0.1 Ablation profile, h a /<h m > Radial position, x/l FIG. 3: An AFM study of a single crater ablated with a 200-fs and 780-nm laser pulse of F 0 = 12.6 J/cm 2. (a) The AFM image. b) The ablation profile at the center-line of the crater. The ablation depth, h a, is normalized by an average melt depth of h m = 1 µm and the radial distance, r, is normalized by a characteristic radial dimension of L = 5 µm. Ben-Yakar et al. 27

28 Incident beam, F 0 plasma AF th AF 0 h a F (z) a melt h m solid z T(z,0) ablation depth T th melt front T m z h a h m FIG. 4: Absorbed laser energy deposition according to the Beer-Lambert law. A sketch of the resulting initial temperature distribution inside the material is shown along with definitions of several variables used in the analysis. Ben-Yakar et al. 28

29 1.5 F heat = 1 J/cm 2 1 h m (µm) F heat = 0.5 J/cm time (µs) FIG. 5: Melt depth variation with time for two different laser fluences using the approximation that % of the incident laser energy remains in the glass as thermal energy. The calculations are performed assuming constant thermophysical properties at an average temperature of 900 K. Ben-Yakar et al. 29

30 p(x) - plasma pressure x z=h(x,t) - interface height h(x,0)=<h > m L plasma liquid x Thin melted layer FIG. 6: Description of parameters used in the thin film model to calculate the evolution of the surface of the melt. Ben-Yakar et al. 30

31 1 P pl H 0.5 S = 0.08 plasma molten glass R FIG. 7: Numerical solutions of pressure-driven melt flow, described by Eq. (22), for two different pressure profiles. The plasma pressures are shown in the top plot where the solid curve is given by P pl (R) = exp( R 10 ), and the dashed curve is P pl (R) = exp( R 2 ). The bottom plot shows the evolution of the free surface for each of the pressure profiles. Ben-Yakar et al. 31

32 1 P pl 0.5 P pl = exp( R 10 ) H S = 0.04 plasma molten glass R FIG. 8: Numerical solutions of pressure-driven melt flow for two different bottom substrate shapes. The bottom plot shows the evolution of the free surface for each of the substrate profiles (solid and dashed curves). Ben-Yakar et al. 32