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1030 Applied Physics B – Lasers and Opticsthe calculation of the thermoelastic stress wave propagation isthe temperature distribution at the end of a single femtosecondlaser pulse that reproduces the free-electron distribution describedby (17). In the following calculations, this temperaturedistribution is characterized by T max , the temperature in ◦ C inthe center of the focal volume. From this temperature distributionthe initial thermoelastic pressure (right after the laserpulse, before the acoustic wave has started to propagate) wascalculated usingp(r) =T∫2 (r)β(T )K(T )T 1dT , (20)where T 1 = 20 ◦ C is the temperature before the laser pulse,and T 2 (r) the temperature of the plasma after the laserpulse, which depends on the location within the focal volume.The temperature dependence of the thermal expansioncoefficient β and the compressibility K was takeninto account, using values for metastable water from [159].The time- and space-dependent pressure distribution p(r, t)due to the relaxation of the initial thermoelastic pressurewas calculated using a k-space (spatial frequency) domainpropagation model [160, 161].Because the heated volume is very small (≈ 0.07 µm 3 )and the region subjected to large tensile stress amplitudesis even smaller (see Fig. 14, below), the presence of inhomogeneousnuclei that could facilitate bubble formation isunlikely. Therefore, we have to consider the tensile strengthof pure water to estimate the bubble-formation threshold infemtosecond optical breakdown. Traditionally, the rupture ofa liquid achieved by tensile stress under isothermal conditionsis called ‘cavitation’ while bubble formation due toheating under isobaric conditions is called ‘boiling’ [162].Such a distinction becomes obscure when targets are bothheated and stretched under conditions of stress confinement.We use the crossing of the ‘kinetic spinodal’ as defined byKiselev [163, 164] as threshold criterion for bubble formation.In the thermodynamic theory of phase transitions, thelocus of states of infinite compressibility (∂p/∂V) T = 0, thespinodal, is considered as a boundary of fluid metastable (superheated)states. Physically, however, the metastable statebecomes short-lived due to statistical fluctuations well beforethe spinodal is reached [159, 165]. The kinetic spinodal is thelocus in the phase diagram where the lifetime of metastablestates becomes shorter than a relaxation time to local equilibrium.If the surface tension is known, the physical boundaryof metastable states in this approach is completely determinedby the equation of state only, i.e. by the equilibrium propertiesof the system [163, 164]. This feature distinguishes the kineticspinodal from the homogeneous nucleation limit derived earlierby Fisher, which depends on the size of the volume underconsideration and the duration of the applied stress [166].Unlike Fisher’s equation, the kinetic spinodal reproduces theshape of the spinodal and is applicable in the entire temperaturerange from room temperature to the critical point. Thenucleation thresholds in cells will probably resemble those inpure water because the biomolecules are too small to serve asboundaries for heterogeneous nucleation.FIGURE 13 (a) Binodal, spinodal, and kinetic spinodal of water as a functionof temperature, calculated with the analytic equation of state of Sauland Wagner [240]. The dotted curve corresponds to the homogeneous nucleationlimit in Fisher’s theory [166], the circles indicate experimental dataof Skripov et al. [159], and the triangles indicate the experimental data ofZheng et al. [167] recalculated in p–T coordinates. Figure reproduced withpermission from Ref. [163]. Copyright 1999 Elsevier Science B.V. (b) Peakcompressive and tensile thermoelastic stresses in the focus center producedby a 800-nm, 100-fs pulse focused into water at NA = 1.3, plotted togetherwith the binodal and the kinetic spinodalFigure 13a presents the saturated liquid/vapor curve (binodal),the spinodal, the kinetic spinodal, and the homogeneousnucleation limit derived by Fisher in a p vs T projection of thethermodynamic phase diagram for water. For comparison, experimentaldata by Skripov et al. on the empirical limit of themetastable region and by Zheng et al. on the tensile strengthof water are also shown [159, 167]. In Fig. 13b, the kineticspinodal is plotted together with the peak compressive andtensile thermoelastic stresses in the focus center that are producedwhen an 800-nm, 100-fs pulse is focused into water atNA = 1.3. The temperature at which the tensile stress curvereaches the kinetic spinodal is defined as the bubble-formationthreshold. For larger laser pulse energies, the kinetic spinodalwill be reached in an increasingly large part of the focalregion.To calculate the dynamics of the cavitation bubble producedafter crossing the kinetic spinodal, first the size ofthe bubble nucleus was determined. It was identified withthe extent of the region in which the negative pressure exceedsthe kinetic spinodal limit p(r, t)
VOGEL et al. Mechanisms of femtosecond laser nanosurgery of cells and tissues 1031as the starting nucleus for the cavitation bubble. The heatedand stretched material within the nucleus commences to expandinstantaneously (within less than 1ps) once the kineticspinodal is reached [168]. As driving force for the expansiononly the negative part of the time-dependent stress in thecenter of the focal volume was considered, because the nucleusdoes not exist before the tensile stress arrives. This isin contrast to simulations of heterogeneous cavitation wherepre-existing gas bubbles interact with a time-varying pressurewave [158]. Only cases were simulated in which the temperaturein the center of the focal volume slightly exceeds thephase-transition threshold and the size of the resulting nucleusis small compared with the focal volume. This justifies usingthe tensile stress amplitude in the center of the undisturbedfocal volume as driving force for the bubble expansion. Forlarger laser pulse energies one would need to consider that thetensile stress is diminished by the rupture of the liquid.After the passage of the tensile stress transient, the vaporpressure p v inside the bubble continues to drive the bubble expansion.The initial vapor pressure is calculated for a temperatureaveraged over all volume elements within the nucleus.During bubble growth, it will drop due to the cooling of the expandingbubble content. This cooling is counteracted by heatdiffusion from the liquid surrounding the bubble. The temperatureof this liquid, on the other hand, drops because of heatdiffusion out of the focal volume as depicted by the dashedlines in Fig. 10. To quantify the temporal evolution of the drivingpressure, we consider two limiting cases:Case 1: Bubble size ≪ focal volume. We assume that thebubble initially expands adiabatically until the average temperatureof the bubble content has fallen to the temperatureof the liquid at the nucleus wall. Afterwards, heat flow fromthe surrounding liquid maintains the temperature of the bubblecontent at the same level as that of the surrounding liquid.The bubble pressure p v (t) thus equals the equilibrium vaporpressure corresponding to the temperature at the nucleus wall,which, for NA = 1.3, decays within about 20 ns to 1/e of itsinitial value (Fig. 10a). Justifications for this hypothesis arethe small bubble size and the fact that the bubble nucleus is notempty but contains material that is initially at liquid densityand has a relatively large heat capacity.Case 2: Bubble size comparable to or larger than the focalvolume. When the size of the nucleus becomes comparable tothe size of the focal volume, the bubble expands more rapidly.In this case, the heated liquid shell surrounding the bubble israpidly thinned, which leads to an accelerated heat dissipationinto adjacent liquid. The heat flow into the bubble is probablysmall compared with the amount of heat contained in the materialwithin the bubble nucleus. Therefore, the entire bubbledynamics is modeled as an adiabatic expansion. As a consequence,the vapor pressure drops much faster than in case 1.In both cases, the ongoing phase transition in the bubble wasneglected to obtain tractable expressions for p v (t).Thissimplificationenabled us to use the Gilmore model to describe thebubble dynamics [158, 169, 170]. The pressure drop is in case1 slower and in case 2 faster than the pressure decay correspondingto the actual phase transitions of the bubble content.Therefore, these cases represent upper and lower limits for theevolution of the actual bubble size after femtosecond opticalbreakdown.6.2 Evolution of the stress distributionThe thermalization time of the energy carried bythe free electrons was assumed to be 10 ps. ForNA = 1.3,λ = 800 nm, and a sound velocity in water of c 0 = 1500 m/s,the acoustic transit time to the periphery of the heated regionwith 168-nm radius is 112 ps. Thus, the dimensionlessthermalization time (thermalization time divided byacoustic relaxation time) is tp ∗ = 0.09, which correspondsto a very high degree of stress confinement. The ‘thermalizationpulse’ was assumed to have a Gaussian temporalshape, with the peak at t = 0. By comparison with anexponential thermalization pulse we found that, for shortdimensionless pulse durations tp ∗ , the pulse shape has littleinfluence on the shape and amplitude of the stresswave.Figure 14 shows the spatial stress distribution in radial andaxial directions for various points in time after the release ofthe laser pulse, and Fig. 15 presents the temporal evolution ofthe stress amplitude in the center of the focal volume. All pressureamplitudes are normalized to the peak compressive stresscreated in the focal volume.FIGURE 14 Stress distribution produced by a single femtosecond pulse of800-nm wavelength focused into water (NA = 1.3), for various times afterthe release of the laser pulse; (a) in radial direction, (b) in axial direction. Thethermalization time of the energy carried by the free electrons was assumedto be 10 ps. The dimensionless thermalization time (thermalization time dividedby acoustic relaxation time) was tp ∗ = 0.09. The ‘thermalization pulse’was assumed to have a Gaussian temporal shape, with the peak at t = 0. Thepressure amplitudes are normalized to the peak compressive stress created inthe focal volume
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