Evaporation kinetics of fused silica were measured up to ≈3000K using CO2 laser heating, while solid-gas phase chemistry of silica was assessed with hydrogen, air, and nitrogen. Enhanced evaporation in hydrogen was attributed to an additional reduction pathway, while oxidizing conditions pushed the reaction backwards. The observed mass transport limitations supported use of a near-equilibrium analysis for interpreting kinetic data. A semi-empirical model of the evaporation kinetics is derived that accounts for heating, gas chemistry and transport properties. The approach described should have application to materials laser processing, and in applications requiring knowledge of thermal decomposition chemistry under extreme temperatures.
© 2012 OSA
Silica plays a critical role in many industrial applications such as raw material in refractory linings, fiber optics, optical substrates and, in general, as a component in devices requiring inertness and toughness. However, the same properties that make silica attractive also make it difficult to process. High temperatures above the glass working point (≈2400 K) are required for molding of fused silica, while very reactive species are needed for its chemical etching [1,2]. Furthermore, many of its processing properties depend exponentially on temperature [3–7]. In particular, silica evaporative etching requires extreme temperatures approaching its boiling point of ≈3000K  to be practical for machining under ambient conditions [9–11]. In applications where localized heating is used for machining glass in air these high temperature requirements are often associated with unwanted increases in residual stresses [12,13], formation of rim structures [10,14], and redeposit defects . A reduction in the treatment temperature required for material removal would therefore improve thermal processing in general since the material changes involved are common to a broad range of materials. One approach, which is evaluated in this study, is to increase the laser-driven vapor pressure of silica decomposition products through the implementation of reactive gases in a flow configuration to assist evaporation.
Although the kinetics of a broad range of solid-gas phase chemistry on the evaporation of silica have been studied [8,15–21], systematic studies near the boiling point of silica are lacking because most containment vessels degrade above ≈2000K . Moreover, in situ measurements become blurred from both high blackbody radiation background and high fluxes of heated material. In this work, laser heating was used as a means to reach surface temperatures up to 3000 K at the gas –solid interface, and the impact of selected gas reactivities on the evaporation kinetics of silica could thus be readily assessed at high temperatures. To determine the effect of a broad range of functional chemistries, air, hydrogen-nitrogen mixture, and pure nitrogen gases were selected to relate evaporation kinetics to an oxidizing, reducing, and inert atmosphere, respectively. For these chemistries, the dependence of the evaporation kinetics on temperature and gas flow rate were derived from measurements of the surface temperature and shape profiles of the silica pits formed when exposed to continuous laser heating and controlled gas flow.
Classical treatments of laser-based evaporation model evaporation kinetics are based on the velocity distribution of escaped species within the Knudsen layer close to a hot surface . However, this type of analysis does not account for the specific chemical reactions that occur from a reacting gas, or any shift in the equilibrium of the evaporation reactions from the presence of a gas phase product. In addition, the impact of convective transport resulting from the net evolution of gas products at the reacting surface  or from forced convection is similarly not accounted for by the Knudsen layer modeling. Here we implement for the first time an analysis for laser-based evaporation that assumes near-equilibrium conditions  within a boundary layer where most of the variation in the species concentration occurs. The equilibrium concentration in the vicinity of the gas-solid interface establishes the driving force for the rate of diffusive transport within the boundary layer before mixing and removal in the bulk of the gas stream. The boundary layer thickness, in turn, depends on the gas properties, flow rate, and flow configuration, and determines the transport kinetics via the mass transport coefficient, hm~D/δ where D is the gas species diffusivity and δ is the boundary layer thickness. Therefore, a predictive model for estimating laser-based evaporation rates in our system is obtained through determination of the hm and equilibrium constants, Kp, from which equilibrium concentrations are calculated. The analysis of the evaporation rate data described here provides a means to probe the role of chemical reaction thermodynamics and transport kinetics at the extreme temperatures reached during laser heating of materials in general.
2. Experimental approach and details
2.1 Experimental approach
A schematic of the laser and setup used to heat and apply gas to the surface of the fused silica sample is shown in Fig. 1A . Figure 1B illustrates representative measurements of the pit depth and surface temperature profile used to derive the temperature-dependent evaporation rates resulting from silica laser heating. Temperature measurements were obtained from infrared imaging of the blackbody radiation emitted during heating. The amount of evaporated silica was estimated from the surface shape profiles obtained by interferometry measurements following treatment of the surface. Finally, gas flow rate levels were adjusted with a flow controller, directing a normal incidence gas jet towards the sample surface through a custom nozzle equipped with a laser window for simultaneous laser heating. Additional details of the measurements and setup are provided below.
2.2 Heating of silica surface
Localized heating and evaporation of the silica surface was achieved with a focused CW (λL = 10.6 μm) CO2 laser beam from a Synrad Firestar V20 (Synrad, Mukilteo, WA, USA), with a maximum output power of 20W and power stability of ≈1% over the duration of the exposure. The laser beam profile was nearly a perfect Gaussian with a 1/e2 diameter of ≈1 mm, as characterized by a beam proler (Pyrocam III, Spiricon). The laser power delivered to the sample was set between 6.5 and 7.2W, while the exposure time (laser turn ON-OFF) was fixed at 5 sec. A far infrared laser was selected because laser energy couples efficiently to a narrow absorption depth of≈3-40 μm at λL , thus surface temperatures in excess of 3000K could readily be achieved. However, evaporation above 3000K produced deeper pits that became more difficult to resolve to get surface profile by interferometry. In addition, the Fresnel reflectivity increases, and hence net laser energy absorption tends to decrease when the aspect ratio of the pit (depth to width) approaches ≈1. In contrast, temperatures between 2100 and 2500 K produced shallower pits for which the effects on pit depth from the thermally-induced densification of silica  became significant. The resulting surface depressions – distinguishable from those due to evaporation - were as deep as 100 nm, or about 10% or more of the total pit depth. Below 2100 K, owing to the exponential dependence of the evaporation on temperature, pit depth was dominated by silica compaction such that contributions due to evaporation could not be determined. For these reasons, the exposure time and range of temperatures in this study were optimized as stated to effectively study silica evaporation around its boiling point (2500 – 3100 K).
2.3 Gas treatment of the silica surface
The samples used in this study were Corning 7980 (type III) fused silica (50 mm diameter, 10 mm thick discs). They were fixed vertically on a stage in ambient air and treated by injecting reactive gases using a nozzle with a 3 mm opening in front. Sample-surface spacing was kept fixed ≈5 mm. The gas jet impinged normal to the surface plane and submerged the treated area well beyond the boundaries of the heated site by displacing the ambient air at the reacting surface before onset of laser heating. The laser beam passes through a transparent ZnSe window mounted on the backside of the nozzle and on to the surface through the nozzle front side opening. Compressed gas cylinders were used as the source for the following four dry gases (<1 ppmv H2O), with trace amounts of impurities (< 3 ppmv): 1) dry air (78% nitrogen, 21% oxygen, 1% trace gases), 2) 100% nitrogen, 3) 5% hydrogen, 95% nitrogen, 4) 5% hydrogen, 95% helium. A calibrated OMEGA gas flow controller (FMA 3400) allowed setting the volumetric flow rate to levels up to 10 L/min. The flow was started at least a minute before laser exposure to insure that all the dead volume was removed from the lines and that surface gas submersion was at steady state.
2.4 Depth profile measurement of the silica surface
As discussed in the Results and Discussion section, the amount of material removed at the center of the laser created pits can be attributed exclusively to the net amount of evaporated material. Therefore, after laser exposure, the surface profiles were measured using a ZYGO white light interferometer (Zygo Corp., Middlefield, CT, USA) to determine the pit depth and thus the amounts of evaporated silica at the laser heated sites. The vertical resolution of the instrument is on the order of 1 nm and the lateral resolution ≈0.5 μm. Measured maximum depths at the center of the pits ranged from 1 to 45 μm; and pit diameters ranged from 200 to 500 μm depending on the peak temperature and gas used.
2.5 Temperature measurement of silica surface
During laser exposure, the sample surface temperature was derived from in situ measurements of the blackbody emission from the heated spot using a calibrated LN-cooled HgCdTe camera operating at 33 fps and equipped with a λp=8.9 μm narrowbandpass cold filter . Although the optical resolution of the camera was ≈100 μm (sampled at 40 μm spacing), it can be shown that the calculated temperature profile established by a 1 mm beam varies quite slowly over the length scales studied here. Thermal imaging in a narrow band in the far infrared (λp) is crucial because it probes the surface emission emanating exclusively from within ≈1 μm of the outer surface of silica, which results in an accurate temperature measurement of the interface rather than a bulk average over a gradient . For the purposes of analyzing the data, the gas at the gas-solid interface was assumed to be rapidly thermalized to the measured temperature of the silica surface. For further details, the 2D surface temperature measurement method used in this study was validated and described elsewhere [26–28].
3. Results and discussion
3.1 Estimate of the evaporation rates from surface profile
The temperature and composition dependent evaporation rate, R(T, Ci), was estimated based on the measurement of the depth profile (Fig. 1B) as it relates to the amount of material removed by evaporation. The accuracy of this approach to derive R depends on the assumption that the depth at a particular location is the result of only the evaporation process, and not the result of flow of molten silica or material expulsion from explosive boiling . At the center of the pit where the pit depth is maximum, that assumption is appropriate for two reasons. First, an estimate of the thermocapillary flow suggests relatively little contribution of flow-displaced silica to the total depth. The flow velocity, vf, normal to the surface of the thermocapillary flow can be roughly approximated by vf = (dγ/dT)ΔT/μ , where dγ/dT represents the rate of change of the surface tension with temperature, ΔT, is the temperature drop from the center of the pit to the edge of the pit (Fig. 1B), and μ is the temperature dependent dynamic viscosity. Thus the calculated contribution of the thermocapillary flow to the total displacement of silica at the center of the pit, vf × Δt, can be shown to contribute no more than 2% of the total pit depth.
In addition, any material removed from drag associated with the gas feed flow should be negligible given the gases relative lack of inertia at atmospheric pressure, and the small superficial velocities involved in this study (<25 m/s). Contributions of vapor-induced shear forces and recoil pressure in shaping laser produced cavities in solids were shown to have a negligible impact on the cavity axial depth produced for the relatively slow evaporation conditions used in this study . Finally, none of the surface profiles displayed any roughening within the pit that would normally occur if explosive boiling had taken place, and experimental irradiances were well below the phase explosion threshold (1011 W/cm2) . Therefore, in attributing the axial depth, d, solely to the evaporation of material at that location it is possible to derive a measure of the temperature dependent evaporation kinetics given by R(Tp) ≈ρ´d/Δt, where ρ is the fused silica density, Δt is related to the laser exposure time, and Tp is the peak temperature measured at the center of the pit (Fig. 1B). Using the center depth, d, is particularly convenient because the location of that spot can easily be found from the surface and temperature spatial profiles. Furthermore, restricting the analysis to that location circumvents any ambiguity arising from the non-uniform heating of the Gaussian-shaped laser beam. For the purposes of calculating evaporation rates, the effective exposure time was taken as Δt = 4sec instead of the fixed experimental exposure of five seconds, since the thermal diffusion time needed to approach peak temperatures with thermal diffusivity D = 8 × 10−7 m2/s  is ≈ = 0.98 sec, where is the beam diameter. The resulting error based on the time-integrated experimental evaporation rates extrapolated to lower temperatures is expected to represent <3% of the bottom pit depth. Therefore small variations, δ, in the effective exposure time will have negligible impact and yield error in the estimated evaporation rates that scale mostly as ~1/(Δt + δ), but this error will cancel out when analyzing the ratio of the evaporation rates discussed later. The stability of the temperature temporal profile was characterized in a previous study of the on-axis temperature measurements using similar laser parameters and samples . Peak temperatures remained within <5% of the final peak temperatures reached right before laser turn off, and increased asymptotically at the rate determined by D and as the heat losses from the sample balance out the heat input from laser heating.
3.2 Role of kinetics and rate controlling parameters
The analysis of the evaporation rate data depends on whether the process is transport limited or under reaction kinetics control. In the transport limited regime the controlling parameters are the mass transfer coefficient (hm) and the reaction equilibrium constant (Kp) . When transport is not limiting, the rate constants for the evaporation and condensation reactions become the controlling parameters. The measurement of the R dependence on gas flow rate is used to distinguish between these two alternatives. A lack of dependence of R on flow rate would indicate that the process of evaporation is not transport limited. In Fig. 2 , a representative measurement at 2880K indicates that there is a weak, but significant, dependence of the R on flow for all gases used. This was observed through the experimental temperature range down to ≈2500K where the evaporation rates are the lowest. We conclude from this result that the process of laser-based evaporation seems to be at least partially limited by species transport under the conditions used in this study and that hm and Kp are both rate controlling parameters for the evaporation process. The apparent transport limitations with pure nitrogen further indicates that it is the transport of the products out of the boundary layer that is limiting the R since no gas phase reactants are present in pure nitrogen and air .
3.3 Evaporation kinetics
Having established the dependence of R on mass transport kinetics, which suggests that the process is near-equilibrium, the dependence of silica evaporation on the gas chemistry was determined from measurements of the R in the presence of air, pure nitrogen, 5% hydrogen in nitrogen, and 5% hydrogen in helium. Figure 3 shows a strong dependence of R on the type of gas used. The reduction of silica by hydrogen produced greater evaporation rates than evaporation in an inert nitrogen atmosphere, and pure nitrogen produced greater rates than evaporation in air, an oxidizing environment. Hydrogen effectively allows a reduction of 100-200K in treatment temperature needed to produce the same evaporation rates compared to ambient air conditions. These results were not unexpected since the main endothermic reactions thought to occur at the experimental temperatures are as follows :
With the realization that these reactions are driving the evaporation, interpretation of the results in Fig. 3 become straightforward. Reaction (1) is the main decomposition reaction in the presence of all gases used. The secondary reaction (2) occurs only when hydrogen is added in the gas mixture, providing an additional pathway for the evaporation of silica, which is confirmed by the increased R in the presence of hydrogen in Fig. 3. The presence of O2 manifests itself in two ways. First, evaporation in air is lowered compared to pure nitrogen due to the presence of O2. Second, the O2 evolved in reaction (1), which increases at higher temperatures, tends to slow evaporation by shifting the equilibrium of reaction (1) backward. This shift is apparent in the pure nitrogen and hydrogen data above T≈2900K where R becomes sub-linear, as indicated by the arrows and dashed lines in Fig. 3. The shift is less apparent in air because of the already elevated amount of O2 present (21%). Thus, in air, the O2 produced in reaction (1) does not alter its concentration significantly as indicated by the linearity of R up to the highest temperatures in Fig. 3. For comparison, the maximum evaporation rates were predicted from the Hertz-Knudsen equation, R = Psat(T)/√2πmkBT, where Psat is the vapor pressure of SiO in reaction (1) , m is the molecular mass, and kB is the Boltzmann constant. Not surprisingly, the rates predicted are 2-3 orders of magnitude greater (Fig. 3) because the Hertz-Knudsen model does not account a priori for the mass transport limitations that prevails in this laser-based evaporation study. Also, the contrasting temperature dependence and slope in Fig. 3 reflect the fact that the Hertz-Knudsen formula is derived from the kinetic theory of gases, which scales the escape velocities as 1/√T, while the temperature dependence of the near-equilibrium described below depends on the thermodynamics of the evaporation process and, to a lesser extent, on the temperature dependence of the transport kinetics.
The mass transport problem was further addressed in a separate experiment carried out in helium instead of nitrogen as the carrier gas but with the same hydrogen fraction of 5%. The magnitude of R in helium was expected to be larger because the gas phase diffusivity in helium is larger than in nitrogen [17,32,33], which should result in a greater hm and R in helium if mass transport is the limiting transport mechanism. An increase in R is indeed observed in Fig. 3, which validates our initial assessment from Fig. 2 that the process of laser-based evaporation in our system is mass transport limited. In the transport limited regime, the molar evaporation rate can then be approximated as a function of the hm and the equilibrium SiO concentration, [SiO]eq. Thus, given a product-free gas feed ,
It is on the basis of Eq. (3) that the remainder of the analysis of R will be performed, first, by focusing on calculating the equilibrium case and, second, by determining the hm for our experiments.
3.4 Calculation of the reaction thermodynamics
Although the evaporation data in Fig. 3 could be interpreted on the basis of chemical equilibria and transport, a quantitative approach involves the calculation of the equilibrium species concentration based on the temperature dependent Kp. Kp is given by the free energy, ΔG° = ΔH°-TΔS°, of reaction (1) and (2): Kpi = exp(-ΔGi°/RcT), where Rc is the gas constant and T the temperature for reaction i. A system of coupled equations can then be derived by treating the overall reaction (1) and (2) as elementary reactions:
The terms in parenthesis represent mole fractions, P is the total pressure taken as 1 atm in our system, ni are the initial species quantity in moles, α and ξ, are the extent of reaction for each reaction, nT represents the total number of moles calculated based on the ni and α and ξ. Standard free energies, ΔG°, are available in the literature  and in thermodynamic databases .
In order to compare the equilibrium concentrations calculated from Eqs. (4)–(5) with the evaporation data in Fig. 3, the data is re-plotted in Fig. 4A as the ratio of the R in each gas. This approach allows us to decouple the mass transport problem from the thermodynamic problem, such that from Eq. (3): R(N2)/R(air) ≈[SiO]eq(in N2)/[SiO]eq(in air). In this way, the data in Fig. 4A can be compared directly to the calculated equilibrium concentrations or, equivalently, mole fractions. The implied approximation that hm(in air) ≈hm(in N2) is reasonable because the gases selected have nitrogen as their main constituent ranging from 80 to 100% N2, thus the transport properties of the gas mixtures are expected to be similar. Figure 4A shows a comparison of the experimental ratio of R for evaporation in nitrogen relative to air with the two curves representing the ratio calculated from Eq. (4) and the ΔG° for reaction (1) reported from two sources [8,34]. The corresponding calculated equilibrium SiO mole fractions are provided in Fig. 4B with predicted values determined from ΔG° in . Although the predictions agree with the data at higher temperatures, at lower temperatures a discrepancy is apparent even when accounting for the gap between the two predictions shown as dashed lines in the figure. However, since the discrepancy is within the spread in the data, and the reported ΔG° represent an extrapolation from data below about 2000K, the ΔG° for reaction (1) was fitted to extract an effective ΔG° (Fig. 4). The resulting ΔH°, ΔS°, and ΔG° are summarized in Table 1 , along with the fitted Arrhenius parameters. The apparent activation energies of ≈100 kcal/mol are comparable to experimental values reported previously under mass transport limited conditions for temperatures <2000K .
The source of the discrepancy is uncertain, and comparison of ΔG° from data in this study below 2000 K are not possible because evaporation rates are too slow to be measured as discussed in section 2.2. For evaporation in pure nitrogen relative to that in air, the RN2(T)/Rair(T) data are lower than the ratio calculated based on the reported free energies. In other words, the experimental evaporation rates in air and nitrogen are closer to each other than expected. The temperature dependence of the discrepancy observed in Fig. 4A is difficult to explain on the basis of, for example, concentration dependent diffusivity, on the non-isothermal conditions related to the Gaussian laser beam heating, or on the possibility of re-deposit material affecting the measured pit depth. However, evaporation rates and the resulting mass transfer limitations increase at higher temperatures; these are also conditions that more closely approximate a local equilibrium. Therefore, it is reasonable to expect that the near-equilibrium-based model represented by Eqs. (3)–(5) should result in better agreement at higher temperatures (Fig. 4A).
With the hydrogen gas mixture, a discrepancy in the experimental versus calculated ratios of R (H2 mixture/pure N2) was also observed (Fig. 5A ). Again, the data was fitted to extract an effective ΔG°for reaction (2) (Table 1) for the temperatures reached between 2600 K and 3000 K. However, unless a fixed hydrogen concentration was used when solving Eqs. (4)–(5), the data could never be fitted irrespective of the ΔG° value used. This point is illustrated in Fig. 5B, where the dashed curve representing the calculated ratio with an “initial” 5% hydrogen mixture, remains persistently below the data points. With an initial 5% hydrogen fraction, the calculated hydrogen concentration equilibrates locally to values between 0.5% and 2.5% for temperatures ranging from 2600 K to 3000 K, respectively. In contrast, with a “fixed”, invariant 5% hydrogen concentration, the data can now be readily fitted. The ΔG° values used to plot the experimental ratio in Fig. 5B were determined from the enthalpy and entropy in Table 1 and the measured temperatures. The need to keep the hydrogen fixed in order to be able to fit the data suggests that the mass transport of hydrogen is fast enough to maintain a bulk hydrogen concentration throughout the gas-solid interface where it is being consumed in reaction (2). This insight is consistent with the previous observation (section 3.2) that the transport of the products, and not the reactants, is rate limiting. Thus a fixed 5% hydrogen concentration, along with the derived ΔG° (Table 1) were used to complete the modeling of R and to calculate the predicted ratio. In contrast with the RN2(T)/Rair(T) ratio in Fig. 4A, the RH2(T)/RN2(T) data are greater than the predictions although, once again, the agreement improves at higher temperature where the conditions for near-equilibrium are more closely approximated as was the case for the air/nitrogen case. The fact that the experimental RH2(T)/RN2(T) ratio is greater than predicted from Eqs. (4)–(5) indicates that the evaporation rates in H2 are greater than expected relative to R in pure N2. A potential explanation is that the current model does not account for the possibility that hydrogen may also be reacting with the oxygen evolving from reaction (1), thereby pushing the reaction forward to produce the greater than expected silica evaporation rates in H2. Hydrogen could thus react both with silica directly (reaction (2)) and with oxygen in a third reaction to form water vapor (H2(g) + ½O(g) ⇔ H2O(g)). The thermal decomposition of silica in reaction (1) would also take place simultaneously. We are currently addressing this scenario analytically and experimentally for future publication.
3.5 Calculation and fitting of the transport kinetics
The analysis above was based on the relative R in each gas. A complete expression of the absolute R requires the determination of not only the equilibrium SiO concentrations (section 3.4), but also of the mass transport kinetics expressed in the hm. Although many semi-empirical correlations can be used to determine hm in the laminar or turbulent regime [36,37], none were found for the nozzle-impinging flow configuration in our experimental setup. Therefore the hm must be extracted from our data by fitting across the two process variables on which R depends: temperature and flow rate. R data in Figs. 2 and 6 were fitted for hm using Eq. (3) written more explicitly now as R(T,V) = hm(T,V)´[SiO]eq(T), where V is the gas feed volumetric flow rate and the [SiO]eq(T) is determined for each gas from the fitted reaction free energies in section 3.4. For this purpose, generalized expressions describing the kinetics of transport using a boundary layer approximation are useful. Typically used are the dimensionless Sherwood number, Sh, which relates Sh to the Reynolds, Re, and the Schmidt number, Sc. Sh is defined such that Sh = hmL/D = f(Sc = μ/ρD, Re = ρVL/μ), where L is a characteristic length (taken as the beam diameter), μ the dynamic viscosity, D the species diffusivity, and ρ the gas density. All the temperature dependent gas properties in the model were calculated from available data and empirical models to extrapolate the viscosity , diffusivity , and from the ideal gas law for density. To simplify the problem, all the gas mixtures were treated as pure nitrogen, since this is the dominant species. The particular form of the empirical expression for hm is given by:Fig. 6 this expression provides a good fit to the data with the corresponding C, m, and n given in Table 1, corresponding to mass transfer coefficients between 0.07 and 0.7 m/s as calculated from Eq. (6) and the experimental flow rates.
With hm determined and the equilibrium concentration calculated from Kp, a predictive model for estimating the laser-based evaporation of silica is now available which accounts for the temperature dependent gas properties, the thermodynamics of the reaction of the gas phase reactant, and the mass transport configuration in the flow system. Whether the additional reaction pathway that involves the gas phase reaction of hydrogen with oxygen is significant in contributing to the evaporation rate of silica is currently being explored to refine the predictions from the model. The experimental method, along with the analysis should be applicable to a broad range of materials exposed to both steady state heating with lasers and to gases with selected reactivities. The critical insight of this study is that the process is mass-transport limited and therefore dependent on the thermodynamics of the reactions through the free energy. The present analysis can also form the basis to interpret evaporation rate data at extreme temperatures, typically only reachable with laser heating. Thus, this approach could potentially enable the derivation of thermodynamic properties of gas-solid phase reactions at extremes temperatures, provided that accurate measurement of the evaporation rates and temperatures are made as described in this study for example. The approach can also provide insights into the mechanism by which specific gases interact with the solid during reactive etching and should improve control of thermal etching processes in general.
The authors would like to thank Dr. Michael Feit for useful discussions on silica evaporation chemistry, and Sherry Baker for her assistance with white-light interferometry measurements. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
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