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Finite element modeling of the mechanical effects of the UV laser ablation of polymer coatings
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Applied Surface Science 254 (2008) 3531–3539 www.elsevier.com/locate/apsusc
Finite element modeling of the mechanical effects of the UV laser ablation of polymer coatings Ioannis N. Koukoulis a, Christopher G. Provatidis a,*, Savas Georgiou b a
Department of Mechanical Engineering, National Technical University of Athens, Zografou Campus, GR-15773 Athens, Greece b Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 71110 Heraklion, Crete, Greece Received 4 July 2007; received in revised form 23 November 2007; accepted 24 November 2007 Available online 3 December 2007
Abstract Laser ablation constitutes the basis of a number of techniques aiming at the processing and diagnosis of polymeric coatings on a variety of substrates. In all these applications, however, the issue is raised about the mechanical effects of the procedure on the substrate integrity. To this end, we employ finite element modeling for simulating the mechanical effects of UV laser ablation on a polymer specimen, with particular emphasis on the structural modifications that may be induced at areas away from the ablation spot. The cylindrical specimen consists of a poly(methylmethacrylate) (PMMA) film on a silica substrate. The analysis shows that stresses of high enough amplitude may propagate to distances far away from the irradiated spot and may induce deleterious mechanical deformations (e.g., cracks or delaminations). The dependence of the distribution of the tensile stresses on the thickness of the two components, as well as on size of the ablation spot area, is examined. Finally, the possibility of growth of pre-existing defects is shown. The results are overall in very good agreement with experimental observations. # 2007 Elsevier B.V. All rights reserved. Keywords: UV laser ablation; Finite element modeling
1. Introduction Laser ablation has evolved into a most important technique for the structuring and microstructuring of materials in a wide range of technological fields and industrial sectors [1]. In particular, UV ablation provides the means of structuring polymers and biopolymers with minimal, if any, deleterious side effects. Thus, UV ablation is extensively employed in microelectronics for drilling, structuring a post-operation removal of polymers, in biology for scaffolding polymer substrates for micro-arrays. On the very same phenomena relies the use of lasers for photorefractive keratectomy as well as for the laser restoration of painted artworks. Ablation is also often employed as the means for producing high-amplitude ultrasound pulses for diagnostic purposes.
* Corresponding author at: Department of Mechanical Engineering, National Technical University of Athens, 9 Iroon Polytechniou Avenue, Zografou Campus, GR-15773 Athens, Greece. Tel.: +30 210 7721520; fax: +30 210 7722347. E-mail address: [email protected] (C.G. Provatidis). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2007.11.041
Given the importance of these applications, the crucial question arises about the nature and extent of any side effects that may compromise the integrity of the substrate in the shortor-long run. The thermal and chemical effects of UVablation on polymers have been investigated in detail and generally appropriate laser irradiation conditions for their minimization have been established. In contrast, the study of the plausible mechanical effects of UV ablation has been rather limited. There are three plausible sources of such impact, namely the expansion of gaseous products formed within the irradiated volume, the thermoelastic expansion of the laser-irradiated volume and the back-momentum exerted by the ejected material. Ablation has been well demonstrated to result in the development of stress waves of even up to 1 GPa amplitude [1–5]. It can be expected that during propagation through the specimen, these high-amplitude waves may induce structural modifications at areas away from the ablation spot. Thus, in contrast to the photochemical effects, which are confined to the laser irradiated area, the photomechanical effects of UV ablation can be much more delocalized. In most cases, structure formation is limited within the irradiated area. There are, however, a few reports of
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modifications induced by UV ablation at areas away from the irradiation spot. For instance, upon ablation of skin, damage to layers 400 mm below the optical penetration depth was detected through optical examination [6]. Similarly, cellular damage and tissue tearing between the lamellar structures have been reported for corneas upon ablation with ArF excimer irradiation [7–9]. In a different type of application, specifically in the laser restoration of artworks, growth of pre-existing defects has been indicated following laser-assisted varnish removal [10,11]. To investigate further this possibility, holographic interferometry has been used for examining the structural modifications effected upon excimer laser (l = 193 nm and 248 nm) ablation of poly(methylmethacrylate) (PMMA) and polystyrene films cast on suprasil substrates [12,13]. Fringe patterns, indicative of local delaminations, were clearly resolved at various regions over the full extent of the substrate. Importantly, no such changes were detected in the irradiation below the ablation threshold, even after extensive irradiation (1000 pulses). It is also interesting that for well-annealed (i.e., initially largely defect-free) samples, defects are minor upon ablation with few laser pulses, but they grow rapidly in size in the subsequent irradiation pulses. Here, we turn to the use of finite element method (FEM) in order to model stress propagation in the substrate upon UV laser ablation and examine to what extent they may result in mechanical modifications. A number of methods, including analytical approaches [1], molecular dynamics simulations [5], finite element methods [14], or even combinations of them [15], have been employed for modeling UV ablation. Generally, in these studies, the emphasis has been on assessing the relative contributions of thermal and photomechanical (stress-generation) processes to the laser-induced material ejection. In contrast, the present study concerns the stress distribution along the radial direction on the area of the PMMA film. We show first that neither heat diffusion away from the laser irradiation area nor thermoelastic stress generation are sufficient to account for structural effects away from the irradiation spot. Instead, the only viable source for effecting structural modifications is through the mechanical pressure that is exerted during ablation and subsequently propagates within the sample. The implications of these findings for the structural modifications that may be effected upon laser ablation of thin polymer films are discussed. 2. Description of the model 2.1. Finite element formulation Stress propagation in continua is described by the general equations of equilibrium governing the linear dynamic
response of a system of finite elements, which is given by [16]: ¨ þ CUðtÞ ˙ þ KUðtÞ ¼ FðtÞ MUðtÞ
(1)
where M, C and K are the mass, damping, and stiffness matrices; F(t) is the vector of externally applied loads that ˙ and U ¨ are the displacement, depends on the time t; and U, U, velocity, and acceleration vectors of the finite element assemblage. If Rayleigh damping is assumed, the damping matrix C is written in the form: C ¼ aM þ bK
(2)
where a and b are the so-called Rayleigh constants. 2.2. Description of geometry and finite element model In this work, Eq. (1) was solved using the commercial software FEM code ANSYS 10.0. A two-dimensional axisymmetric model of the specimens is used, as this is a common geometry of processed samples in applications. For the simulations the specimens have dimensions typical of the samples employed in the holographic experiments [12]—film thickness in the 50–100 mm range, the quartz substrate thickness of 1–2 mm and a radius of 25 mm, was developed. The upper and lower bound were used, leading to four finite element models (Table 1) that consist of about 7000–14,000 axisymmetric four-noded elements and about equal number of nodes, the number depending on the dimensions of the PMMA film as well as of the substrate. For the film of 50 mm and 100 mm thickness, the element divisions were 3 and 6, respectively. Also, for the substrate of 1 mm and 2 mm thickness, the element divisions were 8 and 16, respectively. For both of them, along the radial direction of 25 mm, the element divisions were 625. The aspect ratio of the rectangular elements was 1:2.35 for the PMMA film and 1:3.13 for the silica substrate, values that give good results (a ratio up to 1:4 can be considered reliable). A satisfactory element length should be lmin/10, where lmin represents the shortest wavelength of interest. In the case of irradiation/heating with nanosecond pulses, the generated acoustic waves are of very high bandwidths (up to the frequency of the laser pulse); however, it is the lower wave modes that carry most energy. The time step constitutes a compromise between the requirement for sufficient time resolution and reasonable computational times. A time step shorter than the Tmin/p, where Tmin represents the period of the highest frequency, was considered appropriate [16]. Additionally, two more FEM models based on the geometrical dimensions of model 3 were analysed; model 5 assumes a small elliptical inclusion while model 6 considers a laser spot diameter of double size.
Table 1 Finite element models and their geometrical specifications Model
PMMA film thickness (mm) Silica substrate thickness (mm)
1
2
3
4
5
6
50 1
50 2
100 1
100 2
100 + small void 1
100 + double Ø spot 1
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In all cases reported in this work, we assume a rigid binding between the nodes in the polymer and the ones in the silica, for otherwise delamination at the polymer/suprasil interface behind the ‘‘irradiated area’’ occurs, thus complicating the examination of the stress propagation further away. Nevertheless, this corresponds to the experimental results where indeed for non-annealed (i.e., weakly adhering) films, laser ablation is found to result almost exclusively in the delamination of a circular area around the spot, with reduced defect formation farther away. 2.3. Loading conditions Although the FEM can analyze quite complex problems, some additional simplifying assumptions were made here so as to have a reliable but sufficiently simplified model to permit direct physical understanding of the involved mechanisms. The first simplifying assumption is the neglect of the temperature distribution away from the irradiated spot after a laser pulse. The second one is the neglect of the thermoelastically generated acoustic wave due to the sudden thermal expansion of the spot. The validity of these assumptions is examined in Section 3. Upon laser ablation, a relatively small amount of material (in the systems studied here, 1–7 mm) is ejected, but because of the high velocities, the exerted back-momentum/recoil results in a compressive wave of very high values [17]. For irradiation with typical nanosecond excimer-laser pulses, the pressure depends on the radiant exposure [3], ranging from 104 Pa at fluences close to the ablation threshold to 109 Pa at fluences at which intense plasma is formed; however, in the present examination, a value of 1 GPa was used. For the simulations, a pressure was applied uniformly and perpendicularly on the area of ablation spot (for an irradiated area much larger than the optical penetration depth, the stress waves develop mainly in the axial direction, whereas the stress in the other two directions can be neglected). Boundary restriction of displacement was set on the periphery of the specimen. The typical excimer laser pulse is 30 ns, but most studies indicate that material ejection and thus the exerted recoil continues up to 100 ns and even 1 ms [1]. Various loading schemes have been examined; three of them are presented in Fig. 1, where the rate of the pressure increase is double the rate of the pressure decrease. The actual loading scheme during laser ablation is expected to be much more complex than the ones depicted in Fig. 1, but nevertheless the simplified ones should capture the essential features of the laser-induced pressure. The pressure is assumed to be uniform above a circular laser spot of diameter d = 1 mm while the case of doubling this diameter is also presented (model 6, Table 1). The total time of solution was set 300 ms, time enough to observe the highest values on stresses. 2.4. Material properties For silica, typical static mechanical properties were used [18]; it is clarified that it has a linear and independent from strain-rate response to stresses up to 1 GPa [19]. On the other
Fig. 1. Loading schemes of pressure as a function of time.
hand, for PMMA, as common for thermoplastic materials, the mechanical properties depend sensitively and in a complex manner on the temperature and the loading rate/strain rate [20]. For instance the elastic modulus and the yield stress are reduced by half upon a temperature change from 20 8C to 60 8C [21– 22]. The Young modulus varies from 1720 MPa [23] to 4000 MPa [24] for intermediate strain rates ð˙e ¼ 18:6 s1 Þ, but it reaches a value of 17,000 MPa for compression at high strain rates [25]. The yield stress for compression is reported to be 28 MPa [25] at low strain rates, but 120 MPa at high rates, whereas for tension, it is reported to be around 30 MPa at low strain rate and 70 MPa at high rates [20,24]. Concerning the Poisson ratio, reported values range from 0.28 [26] to 0.39 [27]. The reasons for this variability are related to the influence of the polymerization degree, of the method of production and plausibly of the environmental conditions during sample storage. Molecular weight also affects the fatigue resistance [28,29]. For the simulations, the values assigned to the material properties of PMMA were the averaged ones of those found in literature. Finally, the elastic response of the material is assumed to be linear, which can be justified by the fact that the investigation concerns areas far from the spot, where the strain rates and stresses are relatively low. The typical material properties needed for a mechanical analysis are the elastic modulus (E) and the Poisson ratio (n), however, more are requested for the transient and thermal or thermoelastic analyses. The extra material properties are the density (r), the heat capacity (Cp), the thermal conductivity, the constant of linear thermal expansion and the damping. The used values for the thermal parameters are shown in Table 2. Concerning damping, a literature survey did not reveal any damping data for silica; also, a relevant publication [30] does not use damping in a finite element model. On the other hand, acoustic damping coefficient was reported for PMMA (103 Np/cm at 20 kHz, increasing to 0.88 Np/cm at 10 MHz [31]); damping is estimated to become significant only after 100 cm and few cm for low- and high-frequency (>10 MHz) waves, respectively. Given that the radius of the specimen under consideration is small enough (25 mm), the ablation-induced stress can reflect 100 times within the polymer/substrate before it is dissipated.
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Table 2 Material properties Material property
Young’s modulus, E (GPa) Poisson ratio, n Density, r (kg/m3) Specific heat capacity, Cp (J/kg 8C) Thermal conductivity (W/(m K)) Thermal expansion (mm/(m K)) Damping, b
PMMA
Silica
3.2 0.35 1190 2000 (25–250 8C) 3000 (250 + 8C) 0.22 80 3.5E 10
75 0.17 2200 750 1.38
In order to introduce the above-mentioned attenuation measurements in our six finite element models, it was decided to calculate adequately reliable damping coefficients (cf. Eq. (2)). To this purpose, these experiments were reproduced in the virtual environment of ANSYS 10.0. Due to restrictions of the software used (ANSYS 10.0) related to the existence of two different material sets, the constant a was considered to be zero while the other constant b (cf. Eq. (2)) had to obtain a single value. Therefore, wave propagation through these specimens was accurately represented using another finite element mesh from which the averaged Rayleigh constant b, being equivalent to the measured attenuation (in Ref. [31]) for the first two frequencies of interest (6 MHz and 10 MHz), was estimated by back-calculation as shown in Table 2.
Tðz; tÞ ¼ T 0 þ ðaeff F LASER =C p Þ expðx2e =Dth tÞ, with xe denoting the distance from the irradiatiated area along the radial direction. Estimations by these formula show that heat diffusion towards the bulk and radially is quite slow compared with the time scales of stress propagation. This is confirmed by FEM analyses of the temperature distribution due to heat conduction after a laser pulse. (In these FEM analyses, an initial temperature of 1000 K was assumed at the spot for a depth of 8 mm and then the thermal conduction is solved through time.) Furthermore, since a laser repetition rate of 1 Hz is assumed, the time between successive laser pulses is much smaller than the heat relaxation time ð1=a2eff Dth Þ so that the temperature within the laser irradiated spot and of the sample has returned to the initial value before irradiation with subsequent laser pulse.
3. Results
3.2. Thermal expansion
3.1. Temperature conduction
The pressure rise developed in a substrate upon isochoric heating with a heating (square pulse) can be estimated by ¯ eff F LASER =rkT cV Þð1 eu =uÞ where b¯ is the thermal DP ¼ ðba expansion coefficient, aeff as previously the effective absorption coefficient, CV the heat capacity at constant volume, kT the isothermal compressibility and u = tpulse/tac where tac is the time required for an acoustic wave to traverse the irradiated volume. The factor within the parentheses accounts for the decrease in the pressure due to wave propagation out of the irradiated volume. Based on typical properties for polymers, DP for nanosecond excimer laser pulses is estimated to be 0.1 MPa. This order of magnitude is confirmed by a transient FEM thermoelastic analysis assuming the temperature to rise from a usual room temperature to 1000 K in the time duration of the laser pulse (30 ns). It is noted that if the finite element model did not consider the melt of the PMMA (520 K), the pressure would hardly approach the value of 1 MPa.
A major concern in modeling the results of UV ablation concerns the influence of the high temperature rise within the irradiated area. For PMMA at 248 nm, it is generally accepted that a thermal mechanism dominates [1,32]. Temperatures (at the polymer surface) have been estimated to be 800–1000 K at the end of the laser pulse. Subsequently the substrate temperature along the irradiation axis (assumed perpendicular to the substrate) scales with depth z (from the film surface) and time t as [33]: aeff F LASER Tðz; tÞ ¼ T 0 þ expða2eff Dth tÞ 2Cp pffiffiffiffiffiffiffiffi z expðaeff zÞerfc aeff Dth t pffiffiffiffiffiffiffiffi 2 Dth t pffiffiffiffiffiffiffiffi z þ expðaeff zÞerfc aeff Dth t þ pffiffiffiffiffiffiffiffi 2 Dth t where T0 = 300 K, aeff the effective absorption coefficient (experimentally estimated to be 1000 cm1), F LASER the laser fluence, Cp the heat capacity at constant pressure, and Dth the heat diffusivity. The equation neglects heat losses due to the thermal decomposition of the polymer and any material desorption; for this reason, we have limited simulations to fluences close and below the ablation thresholds. On the other hand, the temperature evolution on the radial direction is simply
3.3. Stress waves Since the influence of the temperature rise can be neglected, the following presentation focuses on the FEM results of the stress propagation due to the mechanical force generated by the back momentum/recoil. Assuming simple elastic wave theory and negligible thermal effects, the impulsive load applied at the top of the specimen close to its center, induces two types of elastic waves, i.e., one
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dilatational (P-wave, c1) of velocity and one shear (S-wave, c2) of velocity for PMMA and for silica C1PMMA = 1839 m/s and C2PMMA = 751 m/s whereas C1Silica = 5848 m/s and C2Silica = 3688 m/s. In addition, Rayleigh waves travel along the free surface and the interface [33]. Moreover, it is well known that for a single P wave incident, two waves, P and SV, are reflected. This phenomenon is referred to as mode conversion. Similarly, for a single SV wave incident, two waves, SV and P, are reflected. In contrast, the SH wave reflects as itself, with no mode conversion, quite analogous to acoustic waves reflects [33]. Within this context, based on the acoustic impedances of the two materials, reflectivity at the polymer/quartz interface is estimated to be 0.6; reflectivity at the free boundaries is assumed to be 1, where the minus sign means incident compressive wave is reflected as a tensile wave. Understandably, the process gets quite complicated and cannot be solved analytically. Furthermore, the applied force causes bending of the substrate, which further modifies the propagating stress waves. Under these circumstances, the application of the finite element method becomes necessary.
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Fig. 3. Radial distribution of max S1 for models 3 and 4 (PMMA film: 100 mm; substrate: 1 and 2 mm, respectively).
Several FE models have been studied in order to address the influence of the polymer and suprasil substrate thickness, as well as of the ablation spot area, on the stress propagation. In addition, a calculation on one model that includes a small void has been performed. Only the results for the first pulse are presented, since further FEM modeling showed no significant dependence of the estimated stress propagation/distribution on the reduction of the PMMA film thickness within the ablated area that occurs with successive laser pulses. In order to present average values, the calculated stresses refer to mid-thickness of the PMMA film. Although a typical diagram of the maximum stress versus time consists of both tensile and compressive values, the analysis (Figs. 2–5 and 7) focuses on the tensile stress, because it is much more efficient
than the compressive one in resulting in material yielding [2]. The value that indicates the tensile stress is the first principal stress (S1) which is calculated at every single point by determining a special set of coordinate system so that the shear stress components vanish and only the normal stresses remain (from which the higher one is the S1) [34]; clearly, the orientation of this system generally changes from point to point. At every point the maximum value of S1 (max S1) through all the time instants was obtained; it is remarkable that in all cases tested the calculated max S1 was found to be positive, a fact that denotes tensile stress. In general, the results for a radius of less than 1 mm are not presented because this area is affected considerably by the concentrated load, as well as is not an issue of this study since delocalized effects are examined. The distribution of max S1 is presented for the first four models in Figs. 2 and 3, which also present the dependence of the max S1 on substrate (suprasil) thickness for two specific thicknesses of the PMMA film. Moreover, Fig. 4 compares models 1 and 3 which have the same substrate thickness but
Fig. 2. Radial distribution of max S1 for models 1 and 2 (PMMA film: 50 mm; substrate: 1 and 2 mm, respectively).
Fig. 4. Influence of the thickness of PMMA film on the radial distribution of max S1 for models 1 and 3 (PMMA film: 50 mm and 100 mm respectively; substrate thickness: 1 mm).
3.4. Overview of finite element results
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Fig. 5. Comparison of radial distribution of max S1 for model 5 and for pressure duration 300 ns (case 1), 600 ns (case 2) and 1000 ns (case 3).
different PMMA film thickness. Also, Fig. 5 illustrates the influence of a pre-existing small void in the PMMA film at a radius of 10 mm, model 5, for the three above-mentioned cases of loading; the small void has an elliptical shape with the major axis equal to 10 mm and the minor to 1 mm. The analyses of model 5 using different loading schemes prescribe the role of the loading duration as well as the role of a pre-existing defect that probably exists in the mass of polymer film. A point at a radius of 10 mm has been chosen as being far enough from the irradiated area. At this point the displacement in the axial direction for the models 1–4 as well as for model 6 is shown in Fig. 6. From this figure, the 1st natural frequency for the models with 1 mm thick substrate is seen to be f 1 = 4240 Hz and for those with 2 mm is f 2 = 8475 Hz. The estimated natural frequencies are in good agreement with the experimental ones [35]. A comparison of the max S1 distribution between the model 6 which has larger spot size and model 3 is presented in Fig. 7 and of the stress raise factor in Fig. 8.
Fig. 6. Displacement of a node at 10 mm from the spot as a function of time for models 1–4 and 6.
Fig. 7. Comparison of radial distribution of max S1 for models 3 and 6 (spot diameter 2 mm).
An energy aspect during loading is presented for the whole model of PMMA film and silica substrate in Fig. 9 as well as separately for each component in Fig. 10. The values are the total mechanical energy (elastic and kinetic) that has been given to the model through the work of the mechanical force. 4. Discussion According to both analytical models and the finite-element analysis, neither heat diffusion nor thermoelastic-generated stress can contribute by any way in structural deformations even at a short distance away from the irradiation spot, nevertheless on the spot they may. Thus, the experimentally observed effects in the holographic experiments on polymers as well as on the tissues must be ascribed to stresses propagating through the irradiated medium as a result of the recoil exerted by the ejecta. Understandably, the geometry of the sample plays an important role on the distribution of the propagating stress. Depending on the relative thickness of the materials and also on their absolute values, there is a maximum superposition of the
Fig. 8. Stress ratio when the diameter spot size is double/single.
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Fig. 9. The total mechanical energy for the whole model at t1 = 200 ns (peak of pressure value) and the t2 = 600 ns (end of loading).
various traveling waves at specific locations, and these locations are expected to be most prone to failure. Close to the irradiation spot, the waveforms exhibit several oscillations due to multiple reflections of the waves. Further away, the amplitudes decrease and oscillations are hardly discernible, as a result of diffraction resulting in the distortion of the stress profile. In the present case, diffraction is due to the finite size of the laseraffected area (A). At distances d beyond the diffraction length defined as Zf = A/(4lac), lac is the wavelength of the acoustic wave, the initially planar acoustic wave starts becoming spherical into a cone with apex angle 2# ¼ 2 arcsinðd=2zf Þ. In the studied samples consisting of a thin layer on a substrate with different mechanical properties, wave propagation deviates somewhat from this simple formula. The models with the same film thickness exhibit a dependence on the substrate thickness as seen in Figs. 2 and 3. In the former figure the differences are quite large though in the latter the differences are concentrated in a radius of 10 mm; further away the results are almost the same. The observation of those results indicates the importance of the substrate thickness and is shown clearly by the aspect of energy. The models with
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thicker substrate gain less energy at the end of loading time, in particular the energy that the substrate will gain is a function of its own thickness (in the range of polymer thickness which is investigated) (Figs. 9 and 10). In the comparison between the models that have the same substrate thickness, i.e., between models 1 and 3 and between 2 and 4, there are similar results. However, because of the dependence on PMMA thickness, the differences in the max S1 distribution (Fig. 3) between the models 3 and 4 are much smaller than models 1 and 2, although they have the same geometrical differences (Fig. 2). This can be justified by the fact of different gain of energy as shown in Figs. 9 and 10. The models with the same film thickness contain the same energy as the pressure takes its peak value but when the application of the load stops the mechanical energy that remains in the specimen is less in the models with 2 mm substrate compared to those with 1 mm substrate. The energy is dispersed to the substrate or reduced due to the negative work of the external force. This energy aspect can give an explanation for the stress results. Those with the same film thickness take the same energy but the substrate thickness is the one that defines how much energy remains at the end of loading. The energy on the substrate at the end of loading depends only on its own thickness. Additionally, the energy at the film at the models 3–4 is almost the same in contrast with the models 1–2 where the energy that remains within the film is a function of the substrate thickness; the film of 50 mm absorbs less mechanical energy than the film of 100 mm (Fig. 9) at the time of the peak value of applied pressure (1 GPa). Concerning the values and the distribution of max S1, we can distinguish two main cases. The first refers to a relatively thick substrate (2 mm: models 2 and 4) where within the thicker film significantly higher values appear. The second refers to a relatively thin substrate (1mm: models 1 and 3) where within the thicker film slightly higher values appear (Fig. 4). The values are somewhat similar for the area with a radius larger than 10 mm. As a rule of thumb, the results show that stress
Fig. 10. The total mechanical energy for the substrate and the film at the t1 = 200 ns (peak of pressure value) and the t2 = 600 ns (end of loading).
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amplitude scales with the film thickness and inversely with the silica thickness. Alterations in geometrical characteristics change the dynamic behavior of the specimen. As shown in Fig. 6, the vertical displacement reflects the movement of the substrate. Indeed, the maximum displacement is observed at 50 ms for the 1 mm silica substrate, and at 25 ms for the 2 mm one, which is four and two times longer, relatively, than the time that takes for the slowest elastic wave to reach at the same point. Understandably, its amplitude is determined by the quartz thickness. The amplitude is 8.5 mm for the models with 1-mm thick substrate and 2.2 mm for those with the 2-mm thickness, as expected from the fact that the initial substrate bending affected by a given force is inverse proportional to its thickness. The stress rise is due to a combination of the stress waves in the film and in the substrate, as well as of the bending of the substrate. It is remarkable that the highest value is observed at time 67 ms, therefore, suggesting that structural modifications may be initiated at times well after the 30 ns excimer laser pulse. Concerning the influence of the irradiated area, expectably, doubling the radius of the ablation spot (all other geometrical parameters kept the same), and correspondingly the exerted force (constant pressure), results in much higher maximum tensile stresses (Fig. 7). Yet, the stress raise factor is not four throughout the sample (as would be expected from a linear static problem) but it varies with the location, evidently due to the different superposition of the traveling waves. For the particular models presented here, the stress ratio (Fig. 8) has a large variation up to a radius of 7 mm around the irradiated spot taking values from 1.5 to 5 and a smaller one, about 3, further away. Although the amplitude of stresses is not four times higher, the vertical displacement is so (e.g., in Fig. 6, 8.5 mm for model 3 versus 33.7 mm for model 6). This difference in the scaling of the displacements versus that of stresses further shows that the vertical displacements are mainly determined by the substrate bending as well as the total force applied. We also observe that the time of loading is essential. The influence is not only at the area of the void but also at the distribution of the stresses values all over the radial direction. The rise of the stress values is much higher between cases 1 and 2 than between cases 2 and 3, which means that the max S1 stress distribution is inversely proportional to the duration of the loading case in a non-linear way. The explanation for this influence is that when the time is long the specimen has the time to move up as the pressure diminishes and the pressure takes back a part of the given energy through the negative mechanical work. In contrast, for short loading times, as the specimen does not have the time to move up, the pressure continues to push it down, resulting thus in the transfer of a higher mechanical work even if it has lower value. Understandably there is a threshold of time reduction where the applied load gives the major of energy and if the time of enforcement gets a lower value will give less due to lack of time in this case. From the standpoint of applications, it is important to know if these stresses propagating into a sample upon laser ablation
can result in damage. Assuming brittle fracture, failure occurs when the S1 reaches the uniaxial tension strength. Since for PMMA, this quantity is less than 15 MPa, it would appear that the stress field remains within safe conditions, at least for a radius larger than 5 mm. Of course, the stress required for failure is sharply decreased in the presence of defects/cracks. A pre-existing defect can result in a region of a very high concentration of stresses and after some loading cycles, a crack of high enough size may develop, as the experimental results indicate [11,12]. The FEM model indicates that the existence of the void inside the PMMA film results in stresses of 31 MPa for case 1, 53 MPa for case 2 and 120 MPa for case 3 (Fig. 5). For case 3, the stresses are high enough to cause crack and crack propagation eventually after some cycles, but even for the other two cases there is a possibility of reaching the yield stress, depending on the mechanical properties of the film. These are in qualitative correspondence with Griffith’s theory of brittle fracture, according to which a planar sharp edge crack in a linear homogeneous, elastic solid generates a stress field pffiffiffiffiffiffiffi ffi around its tip described by the relation sðr; uÞ ¼ ðK I = 2pr Þ f where r and u are the polar coordinates of a reference system with thepcrack in the origin, f is a trigonometric function and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiptip K I ¼ s pa secðpa=wÞ is the stress-intensity factor dependent on geometry and load characteristic (s is the stress in the undisturbed region, a is the crack size and w is the width of the specimen). Fracture will appear when KI KIC where KIC is the fracture toughness of the material, which for PMMA is 1.65 MPa m1/2. Interestingly, the computational findings indicate a rise of the stresses in the polymer close to the edge of the specimen (Figs. 2, 3 and 10) which can be ascribed to the ‘‘accumulation’’ of higher stresses there due to the boundary restrictions. Indeed, in the experiments, a high number of cracks were found to be at the specimen edge, although this was not the usual case for delaminations. Though the position of voids/defects can be easily specified in models, this is not the case experimentally. In fact, for the case of thin polymer coatings/films on quartz, there may be residual stresses upon film casting [36], and given the weak adhesion strength between the two materials, delamination can happen, even in the absence of pre-existing defects. Experimentally, the degree of residual stresses and of defects is controlled by the extent of sample annealing, but of course this is applied to the whole sample and does not provide any control on the localization of the defect/void. Thus, it was considered not worth the effort to try to model the actual spatial distribution of the laser-induced voids/defects. Furthermore, it should be noted that experimentally, defects are observed only after a number of laser pulses. 5. Conclusions A series of FE analyses has shown the importance of the mechanical parameters that influence the area away from the ablation spot. It is shown that the raise in the temperature concerns only the ablation area and for the rest can be disregarded. Another issue is the geometrical parameters which
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determine the bending which are mainly the thickness and the flexural modulus of the silica substrate. The thickness of the PMMA film is also very determinant due to the reliance with the energy that the film gains and is very much affected by the loading scheme. Finally an already existing defect can cause a significant rise in the stresses and drive its expansion. The size of the spot as well as the laser fluence are important because they are the factors which produce the value of total force. It is also important the flexural modulus of the substrate because defines pretty much the total bending. The remaining stresses, partially small adhesion between the two materials or a pre-existence of a small void can help on the appearance of small defects which rise with an additional load through laser ablation. Acknowledgement The work was supported by PENED 2001 administered by the Greek Ministry of Industry. References
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Applied Surface Science 254 (2008) 3531–3539 www.elsevier.com/locate/apsusc
Finite element modeling of the mechanical effects of the UV laser ablation of polymer coatings Ioannis N. Koukoulis a, Christopher G. Provatidis a,*, Savas Georgiou b a
Department of Mechanical Engineering, National Technical University of Athens, Zografou Campus, GR-15773 Athens, Greece b Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 71110 Heraklion, Crete, Greece Received 4 July 2007; received in revised form 23 November 2007; accepted 24 November 2007 Available online 3 December 2007
Abstract Laser ablation constitutes the basis of a number of techniques aiming at the processing and diagnosis of polymeric coatings on a variety of substrates. In all these applications, however, the issue is raised about the mechanical effects of the procedure on the substrate integrity. To this end, we employ finite element modeling for simulating the mechanical effects of UV laser ablation on a polymer specimen, with particular emphasis on the structural modifications that may be induced at areas away from the ablation spot. The cylindrical specimen consists of a poly(methylmethacrylate) (PMMA) film on a silica substrate. The analysis shows that stresses of high enough amplitude may propagate to distances far away from the irradiated spot and may induce deleterious mechanical deformations (e.g., cracks or delaminations). The dependence of the distribution of the tensile stresses on the thickness of the two components, as well as on size of the ablation spot area, is examined. Finally, the possibility of growth of pre-existing defects is shown. The results are overall in very good agreement with experimental observations. # 2007 Elsevier B.V. All rights reserved. Keywords: UV laser ablation; Finite element modeling
1. Introduction Laser ablation has evolved into a most important technique for the structuring and microstructuring of materials in a wide range of technological fields and industrial sectors [1]. In particular, UV ablation provides the means of structuring polymers and biopolymers with minimal, if any, deleterious side effects. Thus, UV ablation is extensively employed in microelectronics for drilling, structuring a post-operation removal of polymers, in biology for scaffolding polymer substrates for micro-arrays. On the very same phenomena relies the use of lasers for photorefractive keratectomy as well as for the laser restoration of painted artworks. Ablation is also often employed as the means for producing high-amplitude ultrasound pulses for diagnostic purposes.
* Corresponding author at: Department of Mechanical Engineering, National Technical University of Athens, 9 Iroon Polytechniou Avenue, Zografou Campus, GR-15773 Athens, Greece. Tel.: +30 210 7721520; fax: +30 210 7722347. E-mail address: [email protected] (C.G. Provatidis). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2007.11.041
Given the importance of these applications, the crucial question arises about the nature and extent of any side effects that may compromise the integrity of the substrate in the shortor-long run. The thermal and chemical effects of UVablation on polymers have been investigated in detail and generally appropriate laser irradiation conditions for their minimization have been established. In contrast, the study of the plausible mechanical effects of UV ablation has been rather limited. There are three plausible sources of such impact, namely the expansion of gaseous products formed within the irradiated volume, the thermoelastic expansion of the laser-irradiated volume and the back-momentum exerted by the ejected material. Ablation has been well demonstrated to result in the development of stress waves of even up to 1 GPa amplitude [1–5]. It can be expected that during propagation through the specimen, these high-amplitude waves may induce structural modifications at areas away from the ablation spot. Thus, in contrast to the photochemical effects, which are confined to the laser irradiated area, the photomechanical effects of UV ablation can be much more delocalized. In most cases, structure formation is limited within the irradiated area. There are, however, a few reports of
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modifications induced by UV ablation at areas away from the irradiation spot. For instance, upon ablation of skin, damage to layers 400 mm below the optical penetration depth was detected through optical examination [6]. Similarly, cellular damage and tissue tearing between the lamellar structures have been reported for corneas upon ablation with ArF excimer irradiation [7–9]. In a different type of application, specifically in the laser restoration of artworks, growth of pre-existing defects has been indicated following laser-assisted varnish removal [10,11]. To investigate further this possibility, holographic interferometry has been used for examining the structural modifications effected upon excimer laser (l = 193 nm and 248 nm) ablation of poly(methylmethacrylate) (PMMA) and polystyrene films cast on suprasil substrates [12,13]. Fringe patterns, indicative of local delaminations, were clearly resolved at various regions over the full extent of the substrate. Importantly, no such changes were detected in the irradiation below the ablation threshold, even after extensive irradiation (1000 pulses). It is also interesting that for well-annealed (i.e., initially largely defect-free) samples, defects are minor upon ablation with few laser pulses, but they grow rapidly in size in the subsequent irradiation pulses. Here, we turn to the use of finite element method (FEM) in order to model stress propagation in the substrate upon UV laser ablation and examine to what extent they may result in mechanical modifications. A number of methods, including analytical approaches [1], molecular dynamics simulations [5], finite element methods [14], or even combinations of them [15], have been employed for modeling UV ablation. Generally, in these studies, the emphasis has been on assessing the relative contributions of thermal and photomechanical (stress-generation) processes to the laser-induced material ejection. In contrast, the present study concerns the stress distribution along the radial direction on the area of the PMMA film. We show first that neither heat diffusion away from the laser irradiation area nor thermoelastic stress generation are sufficient to account for structural effects away from the irradiation spot. Instead, the only viable source for effecting structural modifications is through the mechanical pressure that is exerted during ablation and subsequently propagates within the sample. The implications of these findings for the structural modifications that may be effected upon laser ablation of thin polymer films are discussed. 2. Description of the model 2.1. Finite element formulation Stress propagation in continua is described by the general equations of equilibrium governing the linear dynamic
response of a system of finite elements, which is given by [16]: ¨ þ CUðtÞ ˙ þ KUðtÞ ¼ FðtÞ MUðtÞ
(1)
where M, C and K are the mass, damping, and stiffness matrices; F(t) is the vector of externally applied loads that ˙ and U ¨ are the displacement, depends on the time t; and U, U, velocity, and acceleration vectors of the finite element assemblage. If Rayleigh damping is assumed, the damping matrix C is written in the form: C ¼ aM þ bK
(2)
where a and b are the so-called Rayleigh constants. 2.2. Description of geometry and finite element model In this work, Eq. (1) was solved using the commercial software FEM code ANSYS 10.0. A two-dimensional axisymmetric model of the specimens is used, as this is a common geometry of processed samples in applications. For the simulations the specimens have dimensions typical of the samples employed in the holographic experiments [12]—film thickness in the 50–100 mm range, the quartz substrate thickness of 1–2 mm and a radius of 25 mm, was developed. The upper and lower bound were used, leading to four finite element models (Table 1) that consist of about 7000–14,000 axisymmetric four-noded elements and about equal number of nodes, the number depending on the dimensions of the PMMA film as well as of the substrate. For the film of 50 mm and 100 mm thickness, the element divisions were 3 and 6, respectively. Also, for the substrate of 1 mm and 2 mm thickness, the element divisions were 8 and 16, respectively. For both of them, along the radial direction of 25 mm, the element divisions were 625. The aspect ratio of the rectangular elements was 1:2.35 for the PMMA film and 1:3.13 for the silica substrate, values that give good results (a ratio up to 1:4 can be considered reliable). A satisfactory element length should be lmin/10, where lmin represents the shortest wavelength of interest. In the case of irradiation/heating with nanosecond pulses, the generated acoustic waves are of very high bandwidths (up to the frequency of the laser pulse); however, it is the lower wave modes that carry most energy. The time step constitutes a compromise between the requirement for sufficient time resolution and reasonable computational times. A time step shorter than the Tmin/p, where Tmin represents the period of the highest frequency, was considered appropriate [16]. Additionally, two more FEM models based on the geometrical dimensions of model 3 were analysed; model 5 assumes a small elliptical inclusion while model 6 considers a laser spot diameter of double size.
Table 1 Finite element models and their geometrical specifications Model
PMMA film thickness (mm) Silica substrate thickness (mm)
1
2
3
4
5
6
50 1
50 2
100 1
100 2
100 + small void 1
100 + double Ø spot 1
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In all cases reported in this work, we assume a rigid binding between the nodes in the polymer and the ones in the silica, for otherwise delamination at the polymer/suprasil interface behind the ‘‘irradiated area’’ occurs, thus complicating the examination of the stress propagation further away. Nevertheless, this corresponds to the experimental results where indeed for non-annealed (i.e., weakly adhering) films, laser ablation is found to result almost exclusively in the delamination of a circular area around the spot, with reduced defect formation farther away. 2.3. Loading conditions Although the FEM can analyze quite complex problems, some additional simplifying assumptions were made here so as to have a reliable but sufficiently simplified model to permit direct physical understanding of the involved mechanisms. The first simplifying assumption is the neglect of the temperature distribution away from the irradiated spot after a laser pulse. The second one is the neglect of the thermoelastically generated acoustic wave due to the sudden thermal expansion of the spot. The validity of these assumptions is examined in Section 3. Upon laser ablation, a relatively small amount of material (in the systems studied here, 1–7 mm) is ejected, but because of the high velocities, the exerted back-momentum/recoil results in a compressive wave of very high values [17]. For irradiation with typical nanosecond excimer-laser pulses, the pressure depends on the radiant exposure [3], ranging from 104 Pa at fluences close to the ablation threshold to 109 Pa at fluences at which intense plasma is formed; however, in the present examination, a value of 1 GPa was used. For the simulations, a pressure was applied uniformly and perpendicularly on the area of ablation spot (for an irradiated area much larger than the optical penetration depth, the stress waves develop mainly in the axial direction, whereas the stress in the other two directions can be neglected). Boundary restriction of displacement was set on the periphery of the specimen. The typical excimer laser pulse is 30 ns, but most studies indicate that material ejection and thus the exerted recoil continues up to 100 ns and even 1 ms [1]. Various loading schemes have been examined; three of them are presented in Fig. 1, where the rate of the pressure increase is double the rate of the pressure decrease. The actual loading scheme during laser ablation is expected to be much more complex than the ones depicted in Fig. 1, but nevertheless the simplified ones should capture the essential features of the laser-induced pressure. The pressure is assumed to be uniform above a circular laser spot of diameter d = 1 mm while the case of doubling this diameter is also presented (model 6, Table 1). The total time of solution was set 300 ms, time enough to observe the highest values on stresses. 2.4. Material properties For silica, typical static mechanical properties were used [18]; it is clarified that it has a linear and independent from strain-rate response to stresses up to 1 GPa [19]. On the other
Fig. 1. Loading schemes of pressure as a function of time.
hand, for PMMA, as common for thermoplastic materials, the mechanical properties depend sensitively and in a complex manner on the temperature and the loading rate/strain rate [20]. For instance the elastic modulus and the yield stress are reduced by half upon a temperature change from 20 8C to 60 8C [21– 22]. The Young modulus varies from 1720 MPa [23] to 4000 MPa [24] for intermediate strain rates ð˙e ¼ 18:6 s1 Þ, but it reaches a value of 17,000 MPa for compression at high strain rates [25]. The yield stress for compression is reported to be 28 MPa [25] at low strain rates, but 120 MPa at high rates, whereas for tension, it is reported to be around 30 MPa at low strain rate and 70 MPa at high rates [20,24]. Concerning the Poisson ratio, reported values range from 0.28 [26] to 0.39 [27]. The reasons for this variability are related to the influence of the polymerization degree, of the method of production and plausibly of the environmental conditions during sample storage. Molecular weight also affects the fatigue resistance [28,29]. For the simulations, the values assigned to the material properties of PMMA were the averaged ones of those found in literature. Finally, the elastic response of the material is assumed to be linear, which can be justified by the fact that the investigation concerns areas far from the spot, where the strain rates and stresses are relatively low. The typical material properties needed for a mechanical analysis are the elastic modulus (E) and the Poisson ratio (n), however, more are requested for the transient and thermal or thermoelastic analyses. The extra material properties are the density (r), the heat capacity (Cp), the thermal conductivity, the constant of linear thermal expansion and the damping. The used values for the thermal parameters are shown in Table 2. Concerning damping, a literature survey did not reveal any damping data for silica; also, a relevant publication [30] does not use damping in a finite element model. On the other hand, acoustic damping coefficient was reported for PMMA (103 Np/cm at 20 kHz, increasing to 0.88 Np/cm at 10 MHz [31]); damping is estimated to become significant only after 100 cm and few cm for low- and high-frequency (>10 MHz) waves, respectively. Given that the radius of the specimen under consideration is small enough (25 mm), the ablation-induced stress can reflect 100 times within the polymer/substrate before it is dissipated.
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Table 2 Material properties Material property
Young’s modulus, E (GPa) Poisson ratio, n Density, r (kg/m3) Specific heat capacity, Cp (J/kg 8C) Thermal conductivity (W/(m K)) Thermal expansion (mm/(m K)) Damping, b
PMMA
Silica
3.2 0.35 1190 2000 (25–250 8C) 3000 (250 + 8C) 0.22 80 3.5E 10
75 0.17 2200 750 1.38
In order to introduce the above-mentioned attenuation measurements in our six finite element models, it was decided to calculate adequately reliable damping coefficients (cf. Eq. (2)). To this purpose, these experiments were reproduced in the virtual environment of ANSYS 10.0. Due to restrictions of the software used (ANSYS 10.0) related to the existence of two different material sets, the constant a was considered to be zero while the other constant b (cf. Eq. (2)) had to obtain a single value. Therefore, wave propagation through these specimens was accurately represented using another finite element mesh from which the averaged Rayleigh constant b, being equivalent to the measured attenuation (in Ref. [31]) for the first two frequencies of interest (6 MHz and 10 MHz), was estimated by back-calculation as shown in Table 2.
Tðz; tÞ ¼ T 0 þ ðaeff F LASER =C p Þ expðx2e =Dth tÞ, with xe denoting the distance from the irradiatiated area along the radial direction. Estimations by these formula show that heat diffusion towards the bulk and radially is quite slow compared with the time scales of stress propagation. This is confirmed by FEM analyses of the temperature distribution due to heat conduction after a laser pulse. (In these FEM analyses, an initial temperature of 1000 K was assumed at the spot for a depth of 8 mm and then the thermal conduction is solved through time.) Furthermore, since a laser repetition rate of 1 Hz is assumed, the time between successive laser pulses is much smaller than the heat relaxation time ð1=a2eff Dth Þ so that the temperature within the laser irradiated spot and of the sample has returned to the initial value before irradiation with subsequent laser pulse.
3. Results
3.2. Thermal expansion
3.1. Temperature conduction
The pressure rise developed in a substrate upon isochoric heating with a heating (square pulse) can be estimated by ¯ eff F LASER =rkT cV Þð1 eu =uÞ where b¯ is the thermal DP ¼ ðba expansion coefficient, aeff as previously the effective absorption coefficient, CV the heat capacity at constant volume, kT the isothermal compressibility and u = tpulse/tac where tac is the time required for an acoustic wave to traverse the irradiated volume. The factor within the parentheses accounts for the decrease in the pressure due to wave propagation out of the irradiated volume. Based on typical properties for polymers, DP for nanosecond excimer laser pulses is estimated to be 0.1 MPa. This order of magnitude is confirmed by a transient FEM thermoelastic analysis assuming the temperature to rise from a usual room temperature to 1000 K in the time duration of the laser pulse (30 ns). It is noted that if the finite element model did not consider the melt of the PMMA (520 K), the pressure would hardly approach the value of 1 MPa.
A major concern in modeling the results of UV ablation concerns the influence of the high temperature rise within the irradiated area. For PMMA at 248 nm, it is generally accepted that a thermal mechanism dominates [1,32]. Temperatures (at the polymer surface) have been estimated to be 800–1000 K at the end of the laser pulse. Subsequently the substrate temperature along the irradiation axis (assumed perpendicular to the substrate) scales with depth z (from the film surface) and time t as [33]: aeff F LASER Tðz; tÞ ¼ T 0 þ expða2eff Dth tÞ 2Cp pffiffiffiffiffiffiffiffi z expðaeff zÞerfc aeff Dth t pffiffiffiffiffiffiffiffi 2 Dth t pffiffiffiffiffiffiffiffi z þ expðaeff zÞerfc aeff Dth t þ pffiffiffiffiffiffiffiffi 2 Dth t where T0 = 300 K, aeff the effective absorption coefficient (experimentally estimated to be 1000 cm1), F LASER the laser fluence, Cp the heat capacity at constant pressure, and Dth the heat diffusivity. The equation neglects heat losses due to the thermal decomposition of the polymer and any material desorption; for this reason, we have limited simulations to fluences close and below the ablation thresholds. On the other hand, the temperature evolution on the radial direction is simply
3.3. Stress waves Since the influence of the temperature rise can be neglected, the following presentation focuses on the FEM results of the stress propagation due to the mechanical force generated by the back momentum/recoil. Assuming simple elastic wave theory and negligible thermal effects, the impulsive load applied at the top of the specimen close to its center, induces two types of elastic waves, i.e., one
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dilatational (P-wave, c1) of velocity and one shear (S-wave, c2) of velocity for PMMA and for silica C1PMMA = 1839 m/s and C2PMMA = 751 m/s whereas C1Silica = 5848 m/s and C2Silica = 3688 m/s. In addition, Rayleigh waves travel along the free surface and the interface [33]. Moreover, it is well known that for a single P wave incident, two waves, P and SV, are reflected. This phenomenon is referred to as mode conversion. Similarly, for a single SV wave incident, two waves, SV and P, are reflected. In contrast, the SH wave reflects as itself, with no mode conversion, quite analogous to acoustic waves reflects [33]. Within this context, based on the acoustic impedances of the two materials, reflectivity at the polymer/quartz interface is estimated to be 0.6; reflectivity at the free boundaries is assumed to be 1, where the minus sign means incident compressive wave is reflected as a tensile wave. Understandably, the process gets quite complicated and cannot be solved analytically. Furthermore, the applied force causes bending of the substrate, which further modifies the propagating stress waves. Under these circumstances, the application of the finite element method becomes necessary.
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Fig. 3. Radial distribution of max S1 for models 3 and 4 (PMMA film: 100 mm; substrate: 1 and 2 mm, respectively).
Several FE models have been studied in order to address the influence of the polymer and suprasil substrate thickness, as well as of the ablation spot area, on the stress propagation. In addition, a calculation on one model that includes a small void has been performed. Only the results for the first pulse are presented, since further FEM modeling showed no significant dependence of the estimated stress propagation/distribution on the reduction of the PMMA film thickness within the ablated area that occurs with successive laser pulses. In order to present average values, the calculated stresses refer to mid-thickness of the PMMA film. Although a typical diagram of the maximum stress versus time consists of both tensile and compressive values, the analysis (Figs. 2–5 and 7) focuses on the tensile stress, because it is much more efficient
than the compressive one in resulting in material yielding [2]. The value that indicates the tensile stress is the first principal stress (S1) which is calculated at every single point by determining a special set of coordinate system so that the shear stress components vanish and only the normal stresses remain (from which the higher one is the S1) [34]; clearly, the orientation of this system generally changes from point to point. At every point the maximum value of S1 (max S1) through all the time instants was obtained; it is remarkable that in all cases tested the calculated max S1 was found to be positive, a fact that denotes tensile stress. In general, the results for a radius of less than 1 mm are not presented because this area is affected considerably by the concentrated load, as well as is not an issue of this study since delocalized effects are examined. The distribution of max S1 is presented for the first four models in Figs. 2 and 3, which also present the dependence of the max S1 on substrate (suprasil) thickness for two specific thicknesses of the PMMA film. Moreover, Fig. 4 compares models 1 and 3 which have the same substrate thickness but
Fig. 2. Radial distribution of max S1 for models 1 and 2 (PMMA film: 50 mm; substrate: 1 and 2 mm, respectively).
Fig. 4. Influence of the thickness of PMMA film on the radial distribution of max S1 for models 1 and 3 (PMMA film: 50 mm and 100 mm respectively; substrate thickness: 1 mm).
3.4. Overview of finite element results
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Fig. 5. Comparison of radial distribution of max S1 for model 5 and for pressure duration 300 ns (case 1), 600 ns (case 2) and 1000 ns (case 3).
different PMMA film thickness. Also, Fig. 5 illustrates the influence of a pre-existing small void in the PMMA film at a radius of 10 mm, model 5, for the three above-mentioned cases of loading; the small void has an elliptical shape with the major axis equal to 10 mm and the minor to 1 mm. The analyses of model 5 using different loading schemes prescribe the role of the loading duration as well as the role of a pre-existing defect that probably exists in the mass of polymer film. A point at a radius of 10 mm has been chosen as being far enough from the irradiated area. At this point the displacement in the axial direction for the models 1–4 as well as for model 6 is shown in Fig. 6. From this figure, the 1st natural frequency for the models with 1 mm thick substrate is seen to be f 1 = 4240 Hz and for those with 2 mm is f 2 = 8475 Hz. The estimated natural frequencies are in good agreement with the experimental ones [35]. A comparison of the max S1 distribution between the model 6 which has larger spot size and model 3 is presented in Fig. 7 and of the stress raise factor in Fig. 8.
Fig. 6. Displacement of a node at 10 mm from the spot as a function of time for models 1–4 and 6.
Fig. 7. Comparison of radial distribution of max S1 for models 3 and 6 (spot diameter 2 mm).
An energy aspect during loading is presented for the whole model of PMMA film and silica substrate in Fig. 9 as well as separately for each component in Fig. 10. The values are the total mechanical energy (elastic and kinetic) that has been given to the model through the work of the mechanical force. 4. Discussion According to both analytical models and the finite-element analysis, neither heat diffusion nor thermoelastic-generated stress can contribute by any way in structural deformations even at a short distance away from the irradiation spot, nevertheless on the spot they may. Thus, the experimentally observed effects in the holographic experiments on polymers as well as on the tissues must be ascribed to stresses propagating through the irradiated medium as a result of the recoil exerted by the ejecta. Understandably, the geometry of the sample plays an important role on the distribution of the propagating stress. Depending on the relative thickness of the materials and also on their absolute values, there is a maximum superposition of the
Fig. 8. Stress ratio when the diameter spot size is double/single.
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Fig. 9. The total mechanical energy for the whole model at t1 = 200 ns (peak of pressure value) and the t2 = 600 ns (end of loading).
various traveling waves at specific locations, and these locations are expected to be most prone to failure. Close to the irradiation spot, the waveforms exhibit several oscillations due to multiple reflections of the waves. Further away, the amplitudes decrease and oscillations are hardly discernible, as a result of diffraction resulting in the distortion of the stress profile. In the present case, diffraction is due to the finite size of the laseraffected area (A). At distances d beyond the diffraction length defined as Zf = A/(4lac), lac is the wavelength of the acoustic wave, the initially planar acoustic wave starts becoming spherical into a cone with apex angle 2# ¼ 2 arcsinðd=2zf Þ. In the studied samples consisting of a thin layer on a substrate with different mechanical properties, wave propagation deviates somewhat from this simple formula. The models with the same film thickness exhibit a dependence on the substrate thickness as seen in Figs. 2 and 3. In the former figure the differences are quite large though in the latter the differences are concentrated in a radius of 10 mm; further away the results are almost the same. The observation of those results indicates the importance of the substrate thickness and is shown clearly by the aspect of energy. The models with
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thicker substrate gain less energy at the end of loading time, in particular the energy that the substrate will gain is a function of its own thickness (in the range of polymer thickness which is investigated) (Figs. 9 and 10). In the comparison between the models that have the same substrate thickness, i.e., between models 1 and 3 and between 2 and 4, there are similar results. However, because of the dependence on PMMA thickness, the differences in the max S1 distribution (Fig. 3) between the models 3 and 4 are much smaller than models 1 and 2, although they have the same geometrical differences (Fig. 2). This can be justified by the fact of different gain of energy as shown in Figs. 9 and 10. The models with the same film thickness contain the same energy as the pressure takes its peak value but when the application of the load stops the mechanical energy that remains in the specimen is less in the models with 2 mm substrate compared to those with 1 mm substrate. The energy is dispersed to the substrate or reduced due to the negative work of the external force. This energy aspect can give an explanation for the stress results. Those with the same film thickness take the same energy but the substrate thickness is the one that defines how much energy remains at the end of loading. The energy on the substrate at the end of loading depends only on its own thickness. Additionally, the energy at the film at the models 3–4 is almost the same in contrast with the models 1–2 where the energy that remains within the film is a function of the substrate thickness; the film of 50 mm absorbs less mechanical energy than the film of 100 mm (Fig. 9) at the time of the peak value of applied pressure (1 GPa). Concerning the values and the distribution of max S1, we can distinguish two main cases. The first refers to a relatively thick substrate (2 mm: models 2 and 4) where within the thicker film significantly higher values appear. The second refers to a relatively thin substrate (1mm: models 1 and 3) where within the thicker film slightly higher values appear (Fig. 4). The values are somewhat similar for the area with a radius larger than 10 mm. As a rule of thumb, the results show that stress
Fig. 10. The total mechanical energy for the substrate and the film at the t1 = 200 ns (peak of pressure value) and the t2 = 600 ns (end of loading).
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amplitude scales with the film thickness and inversely with the silica thickness. Alterations in geometrical characteristics change the dynamic behavior of the specimen. As shown in Fig. 6, the vertical displacement reflects the movement of the substrate. Indeed, the maximum displacement is observed at 50 ms for the 1 mm silica substrate, and at 25 ms for the 2 mm one, which is four and two times longer, relatively, than the time that takes for the slowest elastic wave to reach at the same point. Understandably, its amplitude is determined by the quartz thickness. The amplitude is 8.5 mm for the models with 1-mm thick substrate and 2.2 mm for those with the 2-mm thickness, as expected from the fact that the initial substrate bending affected by a given force is inverse proportional to its thickness. The stress rise is due to a combination of the stress waves in the film and in the substrate, as well as of the bending of the substrate. It is remarkable that the highest value is observed at time 67 ms, therefore, suggesting that structural modifications may be initiated at times well after the 30 ns excimer laser pulse. Concerning the influence of the irradiated area, expectably, doubling the radius of the ablation spot (all other geometrical parameters kept the same), and correspondingly the exerted force (constant pressure), results in much higher maximum tensile stresses (Fig. 7). Yet, the stress raise factor is not four throughout the sample (as would be expected from a linear static problem) but it varies with the location, evidently due to the different superposition of the traveling waves. For the particular models presented here, the stress ratio (Fig. 8) has a large variation up to a radius of 7 mm around the irradiated spot taking values from 1.5 to 5 and a smaller one, about 3, further away. Although the amplitude of stresses is not four times higher, the vertical displacement is so (e.g., in Fig. 6, 8.5 mm for model 3 versus 33.7 mm for model 6). This difference in the scaling of the displacements versus that of stresses further shows that the vertical displacements are mainly determined by the substrate bending as well as the total force applied. We also observe that the time of loading is essential. The influence is not only at the area of the void but also at the distribution of the stresses values all over the radial direction. The rise of the stress values is much higher between cases 1 and 2 than between cases 2 and 3, which means that the max S1 stress distribution is inversely proportional to the duration of the loading case in a non-linear way. The explanation for this influence is that when the time is long the specimen has the time to move up as the pressure diminishes and the pressure takes back a part of the given energy through the negative mechanical work. In contrast, for short loading times, as the specimen does not have the time to move up, the pressure continues to push it down, resulting thus in the transfer of a higher mechanical work even if it has lower value. Understandably there is a threshold of time reduction where the applied load gives the major of energy and if the time of enforcement gets a lower value will give less due to lack of time in this case. From the standpoint of applications, it is important to know if these stresses propagating into a sample upon laser ablation
can result in damage. Assuming brittle fracture, failure occurs when the S1 reaches the uniaxial tension strength. Since for PMMA, this quantity is less than 15 MPa, it would appear that the stress field remains within safe conditions, at least for a radius larger than 5 mm. Of course, the stress required for failure is sharply decreased in the presence of defects/cracks. A pre-existing defect can result in a region of a very high concentration of stresses and after some loading cycles, a crack of high enough size may develop, as the experimental results indicate [11,12]. The FEM model indicates that the existence of the void inside the PMMA film results in stresses of 31 MPa for case 1, 53 MPa for case 2 and 120 MPa for case 3 (Fig. 5). For case 3, the stresses are high enough to cause crack and crack propagation eventually after some cycles, but even for the other two cases there is a possibility of reaching the yield stress, depending on the mechanical properties of the film. These are in qualitative correspondence with Griffith’s theory of brittle fracture, according to which a planar sharp edge crack in a linear homogeneous, elastic solid generates a stress field pffiffiffiffiffiffiffi ffi around its tip described by the relation sðr; uÞ ¼ ðK I = 2pr Þ f where r and u are the polar coordinates of a reference system with thepcrack in the origin, f is a trigonometric function and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiptip K I ¼ s pa secðpa=wÞ is the stress-intensity factor dependent on geometry and load characteristic (s is the stress in the undisturbed region, a is the crack size and w is the width of the specimen). Fracture will appear when KI KIC where KIC is the fracture toughness of the material, which for PMMA is 1.65 MPa m1/2. Interestingly, the computational findings indicate a rise of the stresses in the polymer close to the edge of the specimen (Figs. 2, 3 and 10) which can be ascribed to the ‘‘accumulation’’ of higher stresses there due to the boundary restrictions. Indeed, in the experiments, a high number of cracks were found to be at the specimen edge, although this was not the usual case for delaminations. Though the position of voids/defects can be easily specified in models, this is not the case experimentally. In fact, for the case of thin polymer coatings/films on quartz, there may be residual stresses upon film casting [36], and given the weak adhesion strength between the two materials, delamination can happen, even in the absence of pre-existing defects. Experimentally, the degree of residual stresses and of defects is controlled by the extent of sample annealing, but of course this is applied to the whole sample and does not provide any control on the localization of the defect/void. Thus, it was considered not worth the effort to try to model the actual spatial distribution of the laser-induced voids/defects. Furthermore, it should be noted that experimentally, defects are observed only after a number of laser pulses. 5. Conclusions A series of FE analyses has shown the importance of the mechanical parameters that influence the area away from the ablation spot. It is shown that the raise in the temperature concerns only the ablation area and for the rest can be disregarded. Another issue is the geometrical parameters which
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determine the bending which are mainly the thickness and the flexural modulus of the silica substrate. The thickness of the PMMA film is also very determinant due to the reliance with the energy that the film gains and is very much affected by the loading scheme. Finally an already existing defect can cause a significant rise in the stresses and drive its expansion. The size of the spot as well as the laser fluence are important because they are the factors which produce the value of total force. It is also important the flexural modulus of the substrate because defines pretty much the total bending. The remaining stresses, partially small adhesion between the two materials or a pre-existence of a small void can help on the appearance of small defects which rise with an additional load through laser ablation. Acknowledgement The work was supported by PENED 2001 administered by the Greek Ministry of Industry. References
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