Abstract

A new diagnostic approach for assessing the in-depth laser induced modifications upon ultraviolet polymer irradiation is presented. The methodology relies on the observation of morphological alterations in the bulk material (Paraloid B72) by using third harmonic generation. This non destructive methodology allows the detailed and accurate imaging of the structurally laser modified zone extent in the vicinity of the irradiated area. Additionally, for the first time, the visualization and quantitative determination of the contour of the laser-induced swelling/bulk material interface is reported. The observed polymer surface swelling following single-pulse KrF laser irradiation at sub-ablation fluences is interpreted in the context of a model for laser-induced bubble formation due to droplet explosion mechanism.

©2012 Optical Society of America

1. Introduction

Laser ablation of polymers has proven to be important for a great variety of applications, while at the same time has been the subject of extensive scientific research. In fact, many applications have been put on the map due to the characteristic features of UV laser processing, even if the mechanistic aspects underlying polymer ablation are still under investigation. To illuminate the processes taking place upon UV laser irradiation of polymers (mainly with nanosecond laser pulses), a range of studies [13] have examined the dependence of ablation characteristics (threshold, etching efficiency, nature and range of products) on laser irradiation (laser wavelength, fluence and pulse width) and material parameters (absorptivity, molecular weight, chemical structure, etc). The laser-polymer interactions occurring below the ablation threshold regime, as well as the in-depth laser induced changes upon UV polymer irradiation have also been examined [46]. For achieving the optimum result, it is crucial that the initial steps of materials processing by the UV laser beam are monitored and understood. Towards this end, the study of the sub-ablation processes can reveal the dynamics of the laser interactions with polymers that lead to the onset of ablation. Furthermore, the determination of the laser affected zone extent is of great importance since it crucially determines the successful implication of laser ablation in diverse applications. This is precisely the problem that the present work tries to answer; the explicit determination of the laser induced swelling/ bulk material interface.

It is herein demonstrated that nonlinear imaging (THG) can be successfully applied for the clarification of intense bubble-formation phenomena effected upon KrF laser irradiation of Paraloid B72 (PB72) [7]. This technique, being a single beam one and allowing rapid measurements (~1 min) of high resolution (~1 μm), is competitive to the till now employed techniques [8]. Most importantly, the significance of this technique lays on the fact that, for the first time, the contours of the laser induced swelling/bulk material interface are accurately outlined and quantified. The enlightenment of the dependence of the turbid foamed domain localization (swelling) within the polymeric materials on the laser irradiation parameters provides the differentiation between the mechanisms of bubble creation and growth.

2. Experimental

80 μm films of Paraloid B72 (a 70:30 co-polymer of ethyl methacrylate and methyl acrylate with molecular weight ~105 kDa) were investigated. Solutions of the co-polymer (in acetone) were casted on a round glass slide of 35 mm diameter and ~70 μm thickness and dried in air. The polymer films were irradiated with a 30 ns KrF laser (Lambda Physik-LPX 210) emitting at 248 nm with single pulses at sub-ablation fluences. It should be also mentioned that PB72 is transparent in the near UV (340 nm), visible and infrared spectral regions [7].

THG is a nonlinear process, in which three photons of a certain frequency ω are effectively “combined” to form a new photon having the triple fundamental frequency 3ω [9]. THG signal originates as soon as the infrared femtosecond (fs) incident laser beam is focused on an interface between two media having different third-order susceptibility χ(3), refractive index values and/or dispersion [10]. The intensity of THG signal strongly depends on the magnitude of these differences. It is this THG feature that can be utilized for the detection of any structural changes or inhomogeneities (e.g. bubbles) of size comparable to the beam focus [11] in the sample following laser irradiation. In our study a linearly polarized laser beam has been used for the excitation of the samples. However, it has been shown [12] that complementary information related to anisotropy changes of the specimens can be extracted via the detection of THG images, by employing a circularly polarized laser light.

The experimental set-up has been described in detail previously [13]. Briefly, the excitation source used is an Amplitude System t-pulse fs laser oscillator (1028 nm, 1 W, 200 fs, 50 MHz). These laser beam specifications combined with tight focusing (20 x, 0.8 numerical aperture (NA) air immersion objective lens) provide very high efficiency on generating nonlinear optical phenomena. The beam is directed to a modified upright optical microscope with a beam axial waist size of 2 μm. The samples were imaged by a CCD camera. The incident energy per pulse on the sample plane is 0.6 nJ. The samples fit into a motorized xyz translation stage and the line scan is performed by using a pair of galvanometric mirrors. Lab View interface controlled both scanning and data acquisitions. Most THG signals propagate with the laser beam and are collected and collimated by employing a condenser lens. A custom made MATLAB algorithm stores the obtained sequential in-depth line THG measurements in a 2D matrix and determines the position (row) of the maximum THG value in each column. This position represents the interface depth between the structurally modified and non-modified region of the irradiated polymer, since THG signal is strongly enhanced at optical inhomogeneities comparable with the beam focus. Subsequently a new null matrix is created, where the corresponding ‘maximum THG signal’ elements are changed to value equal to unity. This matrix is formed into an image showing the border of the swelling within the axial resolution of the previously described apparatus. In order to compensate the apparent depth distortion for the measurements related to the swelling border towards the bulk material, a correction factor F [14] was considered.

F=11NA2nsns2NA2
where ns is the refractive index of the polymer for the central wavelength of our laser oscillator (1028 nm). This formula is derived by calculating the weighted mean apparent depth for all the rays that are focused by the objective from a zero angle of incidence (axial rays) to an angle a, where nosin(a) = NA (no = 1 for air). This approximation for the calculation of the real depth is more accurate when a high NA objective lens (NA ≥ 0.65) is used, rather than multiplying only by the refractive index ns of the material. Finally, the scanning distance (μm) to number of pixels ratio for each line measurement R is determined and the image is scaled at vertical axis by a factor of F/R in comparison to horizontal axis in order to recover the original aspect ratio of the swelling.

3. Results

The collected THG signals are generated not only from the air/polymer, polymer/glass and glass/air interfaces but also from the air/laser-induced swelling and laser-induced swelling/ bulk material interfaces. In fact, the latter signals are those that define the extent of polymer swelling following irradiation. However, the light scattering from the air/laser-induced swelling interface hinders the beam propagation to the underlying layers obstructing thus the THG signal generation and the definition of the polymer swelling extent. To overcome this problem the sample was reversed and THG signals were collected from each side of the sample. By this configuration, we illustrated both the front (towards the air) and the back (towards the bulk material) border of the polymer swelling. The superimposed images from these measurements for two indicative laser fluences are shown in Fig. 1 . The importance of our results lays on the illustration of the polymer swelling towards the bulk material. By this technique the whole laser affected region in the neighborhood of the irradiated area can be visualized.

 figure: Fig. 1

Fig. 1 THG images from a Paraloid B72 film following single pulse irradiation at 248 nm (illustrating a cross section of the irradiated area). The THG signal from the front and the back border of the polymer swelling are illustrated with green and red color respectively indicating its extent following irradiation at φLASER = 400 (a) and 1550 (b) mJ/cm2. The insets present the corresponding unprocessed (raw) THG images prior to the apparent depth correction. The relative percent difference of THG signal across a single column is in the order of 10 – 25% if we compare the maximum column value with the two adjacent THG values.

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From these images, we measured the distance values of both the front and the back border of the swelling from the polymer surface for a series of laser fluences. The former are in accordance with those measured by conventional profilometry confirming, thus, the accuracy of our technique, while the latter are presented in Fig. 2 as a function of the incident laser fluence following single pulse irradiation at 248 nm. The thermally affected swelled zone measured herein is attributed to cavity formation, on which our discussion relies.

 figure: Fig. 2

Fig. 2 Experimental results of the distance values from the back border of the Paraloid B72 swelling (towards the bulk material) to the polymer surface vs. the incident fluence following single pulse irradiation at 248 nm together with the theoretical curve. Droplet explosion model, initial droplet radius 65 nm, αeff = 1000 cm−1.

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4. Discussion

The cavity formed within the laser irradiated materials can be due to cavitation bubbles [15] created and grown during the rarefaction wave (originated from the reflection of the pressure wave from the free surface of the sample). A detailed approach on KrF excimer laser polymer bubbling is published in [16]. The creation of cavitation bubbles relies on the glass transition temperature (Tg) of the polymer, the excess of the laser heated material temperature over Tg, as well as on the value of the maximal tensile stress generated by the laser pulse. Comparison with the bubbling conditions for other polymers reviewed in [16], and simple estimations show that cavitation bubbles cannot be created by such a mechanism at the experimental threshold fluences taking into account the spectroscopic measured absorption coefficient of PB72, α = 150 cm−1, at 248 nm.

The situation significantly changes if we suggest that the effective absorption coefficient at the irradiation conditions can differ from the above linear value. The concept of an effective absorption coefficient is commonly used in polymer ablation studies [2]. In fact, the nonlinear absorption with almost fixed value at a wide range of laser intensities is observed in doped PMMA irradiated by a KrF excimer laser [3]. As far as it concerns the herein studied polymer, following measurements performed upon KrF irradiation at the studied fluences indicated the effective absorption coefficient for PB72 to be in the range of 1000 cm−1.

As it is discussed in [16] for acoustic initiated bubbling at excimer laser irradiation the optimal value of absorption coefficient is 1000 cm−1. The bubble creation process possesses nucleation, growth, and relaxation steps. According to [16] the threshold of laser bubbling is related to homogeneous nucleation threshold followed from the Zeldovich formula. If we employ the above value of absorption coefficient we easily find that the bubbling criterion derived in [16] is fulfilled. However, at the moment, this theory is not developed enough to explain the growth of generated nuclei. Thus, it is quite difficult to compare its results with the experimental data. Below we consider an alternative to the above cavitation bubble generation mechanisms which can be considered for polymer films fabricated by casting.

The experimental curve of the rare border position vs. φLASER consists of two parts (Fig. 2). At high laser fluences zrear levels off due to significant mass and energy loss (owing to either the volatile elimination and ablation onset [16], or to the self-influence of bubbling on bubbling kinetics [17]). In the following we discuss the first part of the experimental curve suggesting that the growth of the each bubble occurs independently.

Let us consider a liquid droplet of radius r within the polymer matrix. This droplet can be originated from solvent residuals from casting. Upon laser irradiation both the matrix and the droplet are heated; the droplet reaching its boiling temperature. If the pressure of the (liquid) droplet saturated vapour is equal to the surface tension pressure, then the cavity will expand. The evaporating droplet will provide enough gaseous molecules to support the bubble growth. If the growth would proceed up to the complete evaporation of the liquids, then the pressure will change from the saturated pressure to the pressure of the ideal gas with the fixed number of gas molecules within the bubble.

We consider the bubble growth dynamic within the frame of the equation:

drdt=ptrη2ση
which can be obtained from Rayleigh-Plesset equation [18] for the case of small bubbles in high-viscous liquid (in which the inertial terms are neglected). Here pt is the tensile stress, η and σ are the viscosity and the surface tension correspondingly.

The temperature dependence of the surface tension coefficient can be addressed by the Guggenheim expression with critical temperature, σ = σ0(1-T/Tc)11/9, Tc ≈700-1000 K. For polymer melts above Tg, η strongly depends on temperature and the Williams-Landel-Flerry formula η = η0exp(T*/(T-T2)) can be used for this dependence, with T* ≈2069 K, T2 = Tg–ΔT2, ΔT2 ≈51.6 K and according to Avramov’s consideration [19], η0 ≈10−4 erg s/cm3.

Equation (2) was employed by Zeldovich [20] when considering the cavitation nucleation phenomenon. This equation describes the ‘classical trajectory’ neglecting the fluctuations. In Eq. (2) pt = pvpA, where vapour pressure pv is

pv(r,T)=psat(T)pAexp(ΛkBTboilingΛkBT)ifr<rsandpv(r,T)=ρlμlRgT(r0r)1/3ifr>rs

Here r0 is the initial radius of the droplet and rs = r0l Rg T/μl psat(T))1/3 corresponds to its full evaporation. Tboiling is the boiling temperature of the liquid at atmospheric pressure pA, Λ is the latent heat of the liquid evaporation, ρl and μl are the density and molar mass of the liquid, kB is Boltzmann constant and Rg is the universal gas constant. In the case of PB72 the liquid is the solvent acetone with μl = 58.08 g/mol, ρl = 0.79 g/cm3, Tboiling = 329.1 K, Λl = 29.1 kJ/mol.

In our model, we solve Eq. (2) assuming that the gas temperature inside the bubble is equal to the temperature of the surrounding material. That is, in Eq. (3), T = T(z, t) is the solution of Eq. (4)

Tt=DT2Tz2
with initial conditions T(z,0) = Troom + (αφ/cPρ)e-αz and boundary condition ∂T/∂z|z = 0 = 0. The droplet explosion starts when psat(T) = 2σ(T)/r0 + pA. The growing continues until pv(r,T) = 2σ(T)/r + pA and then the radius relaxes. It follows from Eq. (4) with the above initial and boundary conditions, that for each point z>0 the temperature initially increases, then approaches the maximum value Tmax(z) and finally decreases. Calculations show that for different fluences at the rear border of the modified layer, zrear (measured in the experiment and shown in Fig. 2) the following approximate relation holds:

psat(Tmax(zrear))2σ(Tmax(zrear))r0+pA

This means that for different fluences, above the threshold, the position of the rear border of bubbling zone corresponds to the fixed value of maximal temperature Tmax(zrear,φ) ≈const. Experimentally, at ϕ = 0.4 J/cm2, the rear border occurs at distance zrear ≈10µm from the initial surface. For αeff = 1000 cm−1, the corresponding value of Tmax (zrear = 10μm, ϕ = 0.4 J/cm2) is calculated to be 394 K. We designate it as T*. Thus, the equation from which the dependence zrear(φ)can be obtained, reads:

Tmax(zrear,ϕ)T*

Figure 2 shows the corresponding curve together with the experimental data. From (5) the value of the initial droplet radius can be found, which in our case is r0 ≈2σ(T*)/(psat(T*)- pA) ≈ 65 nm.

5. Conclusions

The potential of THG imaging technique for reliably monitoring the presence and extent of the laser induced structural modifications in the bulk polymer following UV laser irradiation is herein demonstrated. The advantage of using nonlinear microscopy over conventional techniques for the determination of the laser induced swelling/bulk material interface contour is also shown. The experimental data can be explained on the assumption that the laser light absorption is addressed by the effective absorption coefficient which significantly differs from the value measured at small intensities. The position of the rear border of the foamed layer, measured by the THG technique, as a function of the laser fluence, corresponds approximately to the fixed value of the maximal temperature. The proposed droplet explosion model of bubble creation satisfactorily fits the experimental data if αeff = 1000 cm−1 and the initial radius of the droplet is about r0 65 nm.

Acknowledgments

Research at IESL-FORTH was supported in part by the EC FP7 projects “LASERLABEUROPE” (No 228334), “CHARISMA” (No 228330) and “FAST-DOT” (No 224338). G. J. T. acknowledges the “HERACLITUS II-University of Crete” funded by the European Social Fund and national resources. N. S. and N. B. thank RFBR (grant 11-02-97053-а) for partial financial support.

References and links

1. D. Bäuerle, Laser Processing and Chemistry (Springer-Verlag, 2000).

2. T. Lippert and J. T. Dickinson, “Chemical and spectroscopic aspects of polymer ablation: special features and novel directions,” Chem. Rev. 103(2), 453–486 (2003). [CrossRef]   [PubMed]  

3. G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006). [CrossRef]  

4. N. Bityurin, “Model for laser swelling of a polymer film,” Appl. Surf. Sci. 255(24), 9851–9855 (2009). [CrossRef]  

5. A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. 103(2), 577–644 (2003). [CrossRef]   [PubMed]  

6. S. Georgiou, “Laser cleaning methodologies of polymer substrates,” Adv. Polym. Sci. 168, 1–49 (2004).

7. P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009). [CrossRef]  

8. N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010). [CrossRef]   [PubMed]  

9. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997). [CrossRef]  

10. M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998). [CrossRef]   [PubMed]  

11. D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. 92(2), 603–612 (2007). [CrossRef]   [PubMed]  

12. N. Olivier, F. Aptel, K. Plamann, M. C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18(5), 5028–5040 (2010). [CrossRef]   [PubMed]  

13. G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010). [CrossRef]   [PubMed]  

14. G. W. White, “Improving the accuracy of vertical measurements under the microscope,” Microscope 18, 51–59 (1970).

15. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003). [CrossRef]   [PubMed]  

16. S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010). [CrossRef]  

17. M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002). [CrossRef]  

18. J. P. Frank and J.-M. Mishel, Fundamentals of Cavitation (Kluwer Ac. Publ., 2004).

19. I. Avramov, “Viscosity in disordered media,” J. Non-Cryst. Solids 351(40-42), 3163–3173 (2005). [CrossRef]  

20. Ya. B. Zeldovich, “Theory of new phase formation. Cavitation,” J. Exp. Theor. Phys. 12, 525–538 (1942).

References

  • View by:

  1. D. Bäuerle, Laser Processing and Chemistry (Springer-Verlag, 2000).
  2. T. Lippert and J. T. Dickinson, “Chemical and spectroscopic aspects of polymer ablation: special features and novel directions,” Chem. Rev. 103(2), 453–486 (2003).
    [Crossref] [PubMed]
  3. G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
    [Crossref]
  4. N. Bityurin, “Model for laser swelling of a polymer film,” Appl. Surf. Sci. 255(24), 9851–9855 (2009).
    [Crossref]
  5. A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. 103(2), 577–644 (2003).
    [Crossref] [PubMed]
  6. S. Georgiou, “Laser cleaning methodologies of polymer substrates,” Adv. Polym. Sci. 168, 1–49 (2004).
  7. P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
    [Crossref]
  8. N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
    [Crossref] [PubMed]
  9. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
    [Crossref]
  10. M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
    [Crossref] [PubMed]
  11. D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. 92(2), 603–612 (2007).
    [Crossref] [PubMed]
  12. N. Olivier, F. Aptel, K. Plamann, M. C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18(5), 5028–5040 (2010).
    [Crossref] [PubMed]
  13. G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
    [Crossref] [PubMed]
  14. G. W. White, “Improving the accuracy of vertical measurements under the microscope,” Microscope 18, 51–59 (1970).
  15. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003).
    [Crossref] [PubMed]
  16. S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
    [Crossref]
  17. M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
    [Crossref]
  18. J. P. Frank and J.-M. Mishel, Fundamentals of Cavitation (Kluwer Ac. Publ., 2004).
  19. I. Avramov, “Viscosity in disordered media,” J. Non-Cryst. Solids 351(40-42), 3163–3173 (2005).
    [Crossref]
  20. Ya. B. Zeldovich, “Theory of new phase formation. Cavitation,” J. Exp. Theor. Phys. 12, 525–538 (1942).

2010 (4)

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M. C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18(5), 5028–5040 (2010).
[Crossref] [PubMed]

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
[Crossref]

2009 (2)

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

N. Bityurin, “Model for laser swelling of a polymer film,” Appl. Surf. Sci. 255(24), 9851–9855 (2009).
[Crossref]

2007 (1)

D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. 92(2), 603–612 (2007).
[Crossref] [PubMed]

2006 (1)

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

2005 (1)

I. Avramov, “Viscosity in disordered media,” J. Non-Cryst. Solids 351(40-42), 3163–3173 (2005).
[Crossref]

2004 (1)

S. Georgiou, “Laser cleaning methodologies of polymer substrates,” Adv. Polym. Sci. 168, 1–49 (2004).

2003 (3)

T. Lippert and J. T. Dickinson, “Chemical and spectroscopic aspects of polymer ablation: special features and novel directions,” Chem. Rev. 103(2), 453–486 (2003).
[Crossref] [PubMed]

A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. 103(2), 577–644 (2003).
[Crossref] [PubMed]

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003).
[Crossref] [PubMed]

2002 (1)

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

1998 (1)

M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
[Crossref] [PubMed]

1997 (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

1970 (1)

G. W. White, “Improving the accuracy of vertical measurements under the microscope,” Microscope 18, 51–59 (1970).

1942 (1)

Ya. B. Zeldovich, “Theory of new phase formation. Cavitation,” J. Exp. Theor. Phys. 12, 525–538 (1942).

Amendt, P. A.

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

Andreotti, A.

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

Aptel, F.

Avramov, I.

I. Avramov, “Viscosity in disordered media,” J. Non-Cryst. Solids 351(40-42), 3163–3173 (2005).
[Crossref]

Barad, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Beaurepaire, E.

N. Olivier, F. Aptel, K. Plamann, M. C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18(5), 5028–5040 (2010).
[Crossref] [PubMed]

D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. 92(2), 603–612 (2007).
[Crossref] [PubMed]

Bityurin, N.

N. Bityurin, “Model for laser swelling of a polymer film,” Appl. Surf. Sci. 255(24), 9851–9855 (2009).
[Crossref]

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

Bizheva, K.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Bounos, G.

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

Brakenhoff, G. J.

M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
[Crossref] [PubMed]

Castillejo, M.

S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
[Crossref]

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

Colombini, P.

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

Débarre, D.

D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. 92(2), 603–612 (2007).
[Crossref] [PubMed]

Dickinson, J. T.

T. Lippert and J. T. Dickinson, “Chemical and spectroscopic aspects of polymer ablation: special features and novel directions,” Chem. Rev. 103(2), 453–486 (2003).
[Crossref] [PubMed]

Dyer, P. E.

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003).
[Crossref] [PubMed]

Eisenberg, H.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Elaboudi, I.

S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
[Crossref]

Filippidis, G.

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

Fotakis, C.

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

Georgiou, S.

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

S. Georgiou, “Laser cleaning methodologies of polymer substrates,” Adv. Polym. Sci. 168, 1–49 (2004).

Glinsky, M. E.

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

Hariri, S.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Horowitz, M.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Hutchings, N.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Hyun, C.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Kaufman, Y.

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

Krmpot, A. J.

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

Lazare, S.

S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
[Crossref]

Lippert, T.

T. Lippert and J. T. Dickinson, “Chemical and spectroscopic aspects of polymer ablation: special features and novel directions,” Chem. Rev. 103(2), 453–486 (2003).
[Crossref] [PubMed]

London, R. A.

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

Moayed, A. A.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Müller, M.

M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
[Crossref] [PubMed]

Nevin, A.

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

Olivier, N.

Paltauf, G.

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003).
[Crossref] [PubMed]

Plamann, K.

Pouli, P.

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

Rebollar, E.

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

Sapir, M.

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

Schanne-Klein, M. C.

Selimis, A.

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

Silberberg, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Simpson, T. L.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Sionkowska, A.

S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
[Crossref]

Sorbara, L.

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

Squier, J.

M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
[Crossref] [PubMed]

Strauss, M.

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

Tavernarakis, N.

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

Tserevelakis, G. J.

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

Venugopalan, V.

A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. 103(2), 577–644 (2003).
[Crossref] [PubMed]

Vlachos, M.

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

Vogel, A.

A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. 103(2), 577–644 (2003).
[Crossref] [PubMed]

White, G. W.

G. W. White, “Improving the accuracy of vertical measurements under the microscope,” Microscope 18, 51–59 (1970).

Wilson, K. R.

M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
[Crossref] [PubMed]

Zeldovich, Ya. B.

Ya. B. Zeldovich, “Theory of new phase formation. Cavitation,” J. Exp. Theor. Phys. 12, 525–538 (1942).

Adv. Polym. Sci. (1)

S. Georgiou, “Laser cleaning methodologies of polymer substrates,” Adv. Polym. Sci. 168, 1–49 (2004).

Appl. Phys. Lett. (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

S. Lazare, I. Elaboudi, M. Castillejo, and A. Sionkowska, “Model properties relevant to laser ablation of moderately absorbing polymers,” Appl. Phys., A Mater. Sci. Process. 101(1), 215–224 (2010).
[Crossref]

Appl. Surf. Sci. (2)

P. Pouli, A. Nevin, A. Andreotti, P. Colombini, S. Georgiou, and C. Fotakis, “Laser assisted removal of synthetic conservation materials from paintings using UV radiation of ns and fs pulse duration: morphological studies on model samples,” Appl. Surf. Sci. 255(9), 4955–4960 (2009).
[Crossref]

N. Bityurin, “Model for laser swelling of a polymer film,” Appl. Surf. Sci. 255(24), 9851–9855 (2009).
[Crossref]

Biophys. J. (1)

D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. 92(2), 603–612 (2007).
[Crossref] [PubMed]

Chem. Rev. (3)

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003).
[Crossref] [PubMed]

A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. 103(2), 577–644 (2003).
[Crossref] [PubMed]

T. Lippert and J. T. Dickinson, “Chemical and spectroscopic aspects of polymer ablation: special features and novel directions,” Chem. Rev. 103(2), 453–486 (2003).
[Crossref] [PubMed]

Invest. Ophthalmol. Vis. Sci. (1)

N. Hutchings, T. L. Simpson, C. Hyun, A. A. Moayed, S. Hariri, L. Sorbara, and K. Bizheva, “Swelling of the human cornea revealed by high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 51(9), 4579–4584 (2010).
[Crossref] [PubMed]

J. Appl. Phys. (2)

G. Bounos, A. Selimis, S. Georgiou, E. Rebollar, M. Castillejo, and N. Bityurin, “Dependence of ultraviolet nanosecond laser polymer ablation on polymer molecular weight: Poly(methyl methacrylate) at 248 nm,” J. Appl. Phys. 100(11), 114323 (2006).
[Crossref]

M. Strauss, Y. Kaufman, M. Sapir, P. A. Amendt, R. A. London, and M. E. Glinsky, “Self-consistent coupling of cavitation bubbles in aqueous systems,” J. Appl. Phys. 91(7), 4720–4725 (2002).
[Crossref]

J. Exp. Theor. Phys. (1)

Ya. B. Zeldovich, “Theory of new phase formation. Cavitation,” J. Exp. Theor. Phys. 12, 525–538 (1942).

J. Microsc. (1)

M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191(3), 266–274 (1998).
[Crossref] [PubMed]

J. Non-Cryst. Solids (1)

I. Avramov, “Viscosity in disordered media,” J. Non-Cryst. Solids 351(40-42), 3163–3173 (2005).
[Crossref]

Micron (1)

G. J. Tserevelakis, G. Filippidis, A. J. Krmpot, M. Vlachos, C. Fotakis, and N. Tavernarakis, “Imaging Caenorhabditis elegans embryogenesis by third-harmonic generation microscopy,” Micron 41(5), 444–447 (2010).
[Crossref] [PubMed]

Microscope (1)

G. W. White, “Improving the accuracy of vertical measurements under the microscope,” Microscope 18, 51–59 (1970).

Opt. Express (1)

Other (2)

J. P. Frank and J.-M. Mishel, Fundamentals of Cavitation (Kluwer Ac. Publ., 2004).

D. Bäuerle, Laser Processing and Chemistry (Springer-Verlag, 2000).

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Figures (2)

Fig. 1
Fig. 1 THG images from a Paraloid B72 film following single pulse irradiation at 248 nm (illustrating a cross section of the irradiated area). The THG signal from the front and the back border of the polymer swelling are illustrated with green and red color respectively indicating its extent following irradiation at φLASER = 400 (a) and 1550 (b) mJ/cm2. The insets present the corresponding unprocessed (raw) THG images prior to the apparent depth correction. The relative percent difference of THG signal across a single column is in the order of 10 – 25% if we compare the maximum column value with the two adjacent THG values.
Fig. 2
Fig. 2 Experimental results of the distance values from the back border of the Paraloid B72 swelling (towards the bulk material) to the polymer surface vs. the incident fluence following single pulse irradiation at 248 nm together with the theoretical curve. Droplet explosion model, initial droplet radius 65 nm, αeff = 1000 cm−1.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

F= 1 1N A 2 n s n s 2 N A 2
dr dt = p t r η 2σ η
p v (r,T)= p sat (T) p A exp( Λ k B T boiling Λ k B T )ifr< r s and p v (r,T)= ρ l μ l R g T ( r 0 r ) 1/3 ifr> r s
T t = D T 2 T z 2
p sat ( T max ( z rear )) 2σ( T max ( z rear )) r 0 + p A
T max ( z rear ,ϕ) T *

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