www.elsevier.com/locate/apsusc

Available online at www.sciencedirect.com

Applied Surface Science 254 (2008) 3531–3539

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, Greeceb 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(methyl-

methacrylate) (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 ultra-

sound 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: cprovat@central.ntua.gr (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 short-

or-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

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–35393532

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 inter-

ferometry 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-genera-

tion) 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

Table 1

Finite element models and their geometrical specifications

Model

1 2 3

PMMA film thickness (mm) 50 50 100

Silica substrate thickness (mm) 1 2 1

response of a system of finite elements, which is given by [16]:

MUðtÞ þ CUðtÞ þKUðtÞ ¼ FðtÞ (1)

where M, C and K are the mass, damping, and stiffness

matrices; F(t) is the vector of externally applied loads that

depends on the time t; and U, U, and U are the displacement,

velocity, and acceleration vectors of the finite element assem-

blage. If Rayleigh damping is assumed, the damping matrix C

is written in the form:

C ¼ aMþ 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.

4 5 6

100 100 + small void 100 + double Ø spot

2 1 1

Fig. 1. Loading schemes of pressure as a function of time.

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–3539 3533

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. Never-

theless, 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

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 ðe ¼ 18:6 s�1Þ, 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

(�10�3 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.

Table 2

Material properties

Material property

PMMA Silica

Young’s modulus, E (GPa) 3.2 75

Poisson ratio, n 0.35 0.17

Density, r (kg/m3) 1190 2200

Specific heat capacity, Cp (J/kg 8C) 2000 (25–250 8C) 3000 (250 + 8C) 750

Thermal conductivity (W/(m K)) 0.22 1.38

Thermal expansion (mm/(m K)) 80

Damping, b 3.5E � 10

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–35393534

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.

3. Results

3.1. Temperature conduction

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]:

Tðz; tÞ ¼ T0 þaeffFLASER

2Cp

expða2effDthtÞ

�expð�aeffzÞerfc

�aeff

ffiffiffiffiffiffiffiffiDthtp

� z

2ffiffiffiffiffiffiffiffiDthtp

�

þ expðaeffzÞerfc

�aeff

ffiffiffiffiffiffiffiffiDthtp

þ z

2ffiffiffiffiffiffiffiffiDthtp

��

where T0 = 300 K, aeff the effective absorption coefficient

(experimentally estimated to be �1000 cm�1), FLASER 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

Tðz; tÞ ¼ T0 þ ðaeffFLASER=C pÞ expð�x2e=DthtÞ, with xe denot-

ing 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 tempera-

ture of 1000 K was assumed at the spot for a depth of 8 mm and

then the thermal conduction is solved through time.) Further-

more, 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=a2effDthÞ 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.

3.2. Thermal expansion

The pressure rise developed in a substrate upon isochoric

heating with a heating (square pulse) can be estimated by

DP ¼ ðbaeffFLASER=rkTcVÞð1� e�u=uÞ where b is the thermal

expansion coefficient, aeff as previously the effective absorp-

tion 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.

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

Fig. 3. Radial distribution of max S1 for models 3 and 4 (PMMA film: 100 mm;

substrate: 1 and 2 mm, respectively).

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–3539 3535

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 C2Sili-

ca = 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, SVand 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. Under-

standably, the process gets quite complicated and cannot be

solved analytically. Furthermore, the applied force causes

bending of the substrate, which further modifies the propagat-

ing stress waves. Under these circumstances, the application of

the finite element method becomes necessary.

3.4. Overview of finite element results

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

Fig. 2. Radial distribution of max S1 for models 1 and 2 (PMMA film: 50 mm;

substrate: 1 and 2 mm, respectively).

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. 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).

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). Fig. 7. Comparison of radial distribution of max S1 for models 3 and 6 (spot

diameter 2 mm).

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–35393536

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 f1 = 4240 Hz

and for those with 2 mm is f2 = 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.

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.

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).

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–3539 3537

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 laser-

affected 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=2zfÞ. 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

Fig. 10. The total mechanical energy for the substrate and the film at the t1

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

= 200 ns (peak of pressure value) and the t2 = 600 ns (end of loading).

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–35393538

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

around its tip described by the relation sðr; uÞ ¼ ðKI=ffiffiffiffiffiffiffiffi2prp

Þ fwhere r and u are the polar coordinates of a reference system

with the crack tip in the origin, f is a trigonometric function and

KI ¼ sffiffiffiffiffiffipap ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

secðpa=wÞp

is the stress-intensity factor depen-

dent 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. Experi-

mentally, 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

I.N. Koukoulis et al. / Applied Surface Science 254 (2008) 3531–3539 3539

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

[1] D. Bauerle, Laser Processing and Chemistry, Springer, Berlin, 2000

(Chapters 12 and 13).

[2] G. Paltauf, P.E. Dyer, Photomechanical processes and effects in ablation,

Chem. Rev. 103 (2003) 487–518.

[3] A.D. Zweig, V. Venugopalan, T.F. Deutsch, Stress generated in polyimide

by excimer-laser irradiation, J. Appl. Phys. 74 (6) (1993) 4181.

[4] S. Siano, R. Pini, R. Salimbeni, Appl. Phys. Lett. 74 (1999) 1233.

[5] L.V. Zhigilei, E. Leveugle, B.J. Garrison, Y.G. Yingling, M.I. Zeifman,

Computer simulations of laser ablation of molecular substrates, Chem.

Rev. 103 (2003) 321–347.

[6] S. Watanabe, T.J. Flotte, D.J. McAuliffe, S.L. Jacques, J. Invest. Dermatol.

90 (1988) 761.

[7] A.G. Doukas, D.J. McAuliffe, T.J. Flotte, Ultrasound Med. Biol. 19 (1993)

137.

[8] S.L. Jacques (Ed.), Proceedings of Laser–Tissue Interaction IX, SPIE

Proc. Ser. 3254SPIE, Washington, 1998, and references therein.

[9] C.B. Gabrielly, E. Pacella, S. Abdolrahimzadeh, F. Regine, R. Mollo,

Ophthalmic Surg. Lasers 30 (1999) 442, and references therein.

[10] S. Georgiou, V. Zafiropulos, D. Anglos, C. Balas, V. Tornari, C. Fotakis,

Appl. Surf. Sci. 127–129 (1998) 738.

[11] V. Tornari, D. Fantidou, V. Zafiropulos, N.A. Vainos, C. Fotakis, SPIE

3411 (1998) 420.

[12] A. Bonarou, L. Antonucci, V. Tornari, S. Georgiou, C. Fotakis, Holo-

graphic interferometry for the structural diagnostics of UV laser ablation

of polymer substrates, Appl. Phys. A 73 (2001) 647–651.

[13] A. Athanassiou, E. Andreou, A. Bonarou, V. Tornari, D. Anglos, S.

Georgiou, C. Fotakis, Examination of chemical and structural modifica-

tions in the UV ablation of polymers, Appl. Surf. Sci. 197–198 (2002)

757–763.

[14] J.C. Conde, F. Lusquinos, P. Gonzalez, J. Serra, B. Leon, A. Dima, L.

Cultrera, D. Guido, A. Zocco, A. Perrone, Finite element analysis of the

initial stages of the laser ablation process, Thin Solid Films 453–454

(2004) 323–327.

[15] J.A. Smirnova, L.V. Zhigilei, B.J. Garrison, A combined molecular

dynamics and finite element method technique applied to laser induced

pressure wave propagation, Comput. Phys. Commun. 118 (1999) 11–16.

[16] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-

Hall, NJ, 1982.

[17] P.E. Dyer, R. Srinivasan, Nanosecond photoacoustic studies on ultraviolet

laser ablation of organic polymers, Appl. Phys. Lett. 48 (6) (1986) 445–

447.

[18] http://www.matweb.com.

[19] R. Ochoa, T.P. Swiler, J.H. Simmons, Molecular dynamics studies of

brittle failure in silica: effect of thermal vibrations, J. Non-Cryst. Solids

128 (1991) 57–68.

[20] W. Chen, F. Lu, M. Cheng, Tension and compression tests of two polymers

under quasistatic and dynamic loading, Polym. Test. 21 (2002) 113–121.

[21] V. Jardret, P. Morel, Viscoelastic effects on the scratch resistance of

polymers: relationship between mechanical properties and scratch proper-

ties at various temperatures, Prog. Org. Coat. 48 (2003) 322–331.

[22] E.M. Arruda, M.C. Boyce, R. Jayachandran, Effects of strain rate,

temperature and thermomechanical coupling on the finite strain deforma-

tion of glassy polymers, Mech. Mater. 19 (1995) 193–212.

[23] Q.H. Shah, H. Homma, Fracture criterion of materials subjected to

temperature gradient, Int. J. Pres. Ves. Pip. 62 (1995).

[24] H. Wu, G. Ma, Y. Xia, Experimental study of tensile properties of PMMA

at intermediate strain rate, Mater. Lett. 58 (2004) 3681–3685.

[25] Z. Li, J. Lambros, Strain rate effects on the thermomechanical behavior of

polymers, Int. J. Solids Struct. 38 (20) (2001) 3549–3562.

[26] X.F. Yao, W. Xu, M.Q. Xu, K. Arakawa, T. Mada, K. Takahashi, Experi-

mental study of dynamic fracture behavior of PMMA with overlapping

offset-parallel cracks, Polym. Test. 22 (2003) 663–670.

[27] H. Wada, M. Seika, T.C. Kennedy, C.A. Calder, Investigation of loading

rate and plate thickness effects on dynamic fracture toughness of PMMA,

Eng. Fract. Mech. 54 (6) (1996) 805–811.

[28] R.P. Kusy, A.R. Greenberg, Influence of molecular weight on the dynamic

mechanical properties of poly(methyl methacrylate), J. Therm. Anal. 18

(1980) 117–126.

[29] P. Prentice, Influence of molecular weight on the fracture of poly(methyl

methacrylate) (PMMA), Polymer 24 (1983) 334–350.

[30] T.D. Bennett, L. Li, Modeling laser texturing of silicate glass, J. Appl.

Phys. 89 (2) (2001) 942–950.

[31] J.R. Assay, D.L. Lamberson, A.H. Guenther, Pressure and temperature

dependence of the acoustic velocities in polymethacrylate, J. Appl. Phys.

40 (4) (1969) 1768–1783.

[32] G. Bounos, A. Kolloch, T. Stergiannakos, E. Varatsikou, S. Georgiou,

Assessment of the thermal and structural changes in the nanosecond

irradiation of doped PMMA and PS at 308 nm, 248 nm and 193 nm via the

examination of dopant-deriving product formation, J. Appl. Phys. 98

(2005) 084317.

[33] K.F. Graff, Wave motion in elastic solids, Dover, New York, 1975.

[34] L.E. Malvern, Introduction to the mechanics of a continuous medium,

Pentice-Hall, Inc., Engelwood, NJ, 1969.

[35] E. Esposito, L. Scalise, V. Tornari, Measurement of stress waves in

polymers generated by UV laser ablation, Opt. Lasers Eng. 38 (2002)

207–215.

[36] I. McCulloch, H.T. Man, K. Song, H. Yoon, Mechanical failure in thin-film

nonlinear optical polymers: structure and processing issues, J. Appl.

Polym. Sci. 53 (5) (1994) 665–676.