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Monitoring photochemical reactions using Marangoni flows Joost Muller, Hubertus M.J.M. Wedershoven, and Anton A Darhuber Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00278 • Publication Date (Web): 20 Mar 2017 Downloaded from http://pubs.acs.org on March 28, 2017
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Langmuir
Monitoring photochemical reactions using Marangoni flows J. Muller, H. M. J. M. Wedershoven and A. A. Darhuber Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands (Dated: March 17, 2017) We evaluated the sensitivity and time-resolution of a technique for photochemical reaction monitoring based on interferometric detection of the deformation of liquid films. The reaction products change the local surface tension and induce a Marangoni flow in the liquid film. As a model system, we consider the irradiation of the aliphatic hydrocarbon squalane with broadband deep-UV light. We developed a numerical model that quantitatively reproduces the flow patterns observed in the experiments. Moreover, we present self-similarity solutions that elucidate the mechanisms governing different stages of the dynamics and their parametric dependence. Surface tension changes as small as ∆γ = 10−6 N/m can be detected and time-resolutions below one second can be achieved. Keywords: photochemical reactions, thin liquid films, solutocapillary flow, Marangoni flow, self-similar solutions
I.
INTRODUCTION
Photochemical reactions induced by irradiation with ultraviolet (UV) light are utilized in a large variety of applications such as surface cleaning [1–8], antiviral and bactericidal surface disinfection [9], air and water purification [10–14], photochemical surface modification [15–23], wettability alteration [24–31], material deposition [32–38], surface charging [39–41], photolithography [42, 43], and the patterning of self-assembled monolayers [44–50]. The photopatterning of polymer layers due to photoisomerization reactions of azobenzene groups has been studied intensively [51–65]. Chowdhury et al. [66] reported on experiments regarding UV-induced photopolymerization reactions of aniline films floating on water. Golovin and Volpert [67, 68] studied the Marangoni instability of a thin liquid layer subject to uniform illumination that triggers a reversible photochemical reaction. They performed a linear stability analysis as well as numerical simulations of the evolution equations for the height- and reactant concentration profiles. Dun et al. studied the laser-pulse induced deformation of 50 nm thick Sb2 Te3 films due to a competition between thermocapillary and solutocapillary effects [70]. A displacement of material out of the laser spot was observed for low powers, whereas material accumulation in the laser spot occurred for higher powers. Verma et al. irradiated polystyrene and polymethylmethacrylate layers with intense near-UV radiation at 355 nm, which caused local melting of the polymer. Using various apertures they were able to induce relief patterns in the polymer films with a single laser pulse (≈ 100 ns) [69]. Katzenstein et al. studied pattern formation after UV-illumination of a solid polymer through a mask and subsequent heating above the glass transition temperature [71, 72]. Arshad et al. presented a perturbation analysis of a coupled convection-diffusion model for UV-induced polymer surface deformations and achieved good agreement with experimental results using polystyrene layers [73]. Kim et al. used a photosensitizer to enhance the UV-
photopatterning effect and concluded that an endoperoxide of the photosensitizer is a critical reaction intermediate [74]. Adding a photo-base generator to a polymer film, Kim et al. demonstrated that UV-induced material redistribution could be directed either towards the irradiated or the non-illuminated regions depending on the mode of exposure using either low-intensity and wavelength-selective or high-intensity and broad-band illumination. The authors identified decarboxylation and dehydrogenation reactions, respectively, as the chemical processes driving the flow [75]. In this manuscript, we report on the irradiation of aliphatic, liquid hydrocarbon films on stationary substrates with deep UV light through a mask [see Fig. 1(a)]. Due to photochemical reactions, the chemical composition of the liquid and thus its surface tension is al-
UV-light
x
(b) y
Mask
Liquid film
h0 Substrate
(a)
Camera
Minima
Maximum
FIG. 1. (a) Deep-UV radiation from a deuterium lamp is partially blocked by a metal mask and partially transmitted to an aliphatic liquid hydrocarbon film. Due to photochemical reactions, the surface tension increases, which induces flow towards the illuminated regions and the formation of a film thickness maximum underneath the unmasked areas. (b) Example of thin liquid film deformation induced by deepUV irradiation through a rectangular aperture of dimensions 0.9 × 10 mm (indicated by the red dashed line). The light and dark optical interference fringes allow monitoring of the film thickness distribution. Initial film thickness h0 = 4.8 µm. Image height 4.5 mm.
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tered and becomes spatially non-uniform. As a consequence, so-called solutocapillary Marangoni flows [76–98] are induced that redistribute the liquid film. We demonstrate that measuring the liquid deformation is a sensitive means of monitoring minute composition changes. We developed a numerical model that quantitatively reproduces the morphological and dynamical features observed in the experiments. Moreover, we determined self-similar solutions that illustrate the mechanisms governing different stages of the dynamics and their parametric dependence.
II.
EXPERIMENTAL METHODS
We deposited a thin liquid film of the essentially nonvolatile liquid squalane (purity 99%, Aldrich, product number 234311) onto flat and rigid substrates using spincoating (Brewer Scientific, model CEE200). The initial film thickness was varied in a range from 2 to 5 µm. The substrates consisted of sapphire wafers (Precision Micro-optics, product number PWSP-113112) of thickness dsub = 430 µm and diameter of 5 cm. Sapphire is optically transparent and exhibits a high thermal conductivity k ≈ 25 W/(mK), which helps suppress the potential occurrence of temperature gradients that might mask the effects of concentration gradients. The viscosity of squalane at 23◦ C is µliq = 31.9 mPa s, the surface tension [99] γ = 28.15 mN/m, the density [100] ρ = 805 kg/m3 and the refractive index is nD = 1.452. The self-diffusion coefficient of squalane [101] is Ds ≈ 3 · 10−11 m2 /s. Kowert and Watson studied the diffusion of organic solutes in squalane [102]. The deep-UV irradiation was performed with a 30 W deuterium lamp (Newport, model number 63163). The effective source diameter is approximately 0.5 mm and is located approximately 20 mm above the squalane film. The exit window is made from synthetic quartz in order to minimize absorption of deep-UV radiation. The effective emission wavelengths range approximately from 160 nm to 400 nm. The lamp output does not exhibit any significant start-up effects for illumination times less than one minute. A steady-state value of the intensity is reached within 0.25 s of switching it on, after which the intensity remains constant to within ±2%. The experiments were performed either in an air environment, or a chamber flushed with either dry nitrogen or Ar gas. The aperture consisted of a rectangular slit approximately 1 mm wide and 10 mm long in a 2 mm thick Al plate. The separation between the underside of the Al plate and the squalane film was approximately 0.5 mm. The effective pathlength of the deep-UV light from the exit window of the lamp to the surface of the squalane film was approximately 7.5 mm. The illumination time tUV was varied between 10 and 60 s. A video of a typical experiment is part of the supporting information. For live visualization of the time evolution of the liquid height profile h(x, y, t) during and after UV expo-
sure, we used optical interferometry in a transmission geometry. A high-speed camera (Photron SA-4) fitted with protective optical filters, a lens system and a microscope objective (Leitz Wetzlar NPL 5×, NA = 0.09) was mounted underneath the substrate. Since the camera was blinded by the broadband illumination, we introduced a chopper (Thorlabs, model number MC2000) with its blade (Thorlabs, model number MC1F2) rotating at a frequency of 2 rps. When the illumination was blocked by the blade, interferometry images were recorded using an LED light source (Roithner, center wavelength λ = 625 nm). Figure 1(b) shows an example of an interference image. The height difference represented by two consecutive constructive or destructive interference fringes corresponds to ∆Hint = λ/(2nD ) ≈ 215 nm. The total film thickness modulation is then determined by evaluating the number of interference fringes between the maximum and minimum positions indicated by the solid lines in 1(b) and multiplication by ∆Hint . Interferometry using a single wavelength allows only to determine changes in film thickness, but not absolute values. Therefore, we measured the initial film thickness h0 using a home-built spectral reflectance system comprising a spectrometer (Ocean Optics, model USB4000) and a tungsten-halogen broadband lightsource (Ocean Optics, model LS-1-LL). III.
THEORETICAL MODEL
It is likely that a number of different chemical species result from the reaction. In the following, however, we account for the photogenerated species by means of a single concentration variable c and assume that surface tension has a linear dependence γ(c) = γ0 +
∂γ ∂c
c,
(1)
with a constant coefficient ∂γ/∂c > 0. This is typically a realistic assumption for non-aqueous systems as long as the total photochemical conversion is low. Furthermore, we assume that the photoreaction leads to an increase in the photoproduct concentration at a constant rate proportional to the local light intensity. This may occur e.g. at the liquid-air interface z = h, if the absorption depth is small α−1 ≪ h0 , or homogeneously throughout the thickness of the thin liquid film, if α−1 ≫ h0 . Here, α(λ) is the absorption coefficient of the photochemically-active UV radiation. The timescale of the experiments ranges typically from 10 to 600 s, which by far exceed the diffusive time scale h20 /Ds . 0.8 s. Consequently, in both cases the concentration distribution can be assumed to be vertically equilibrated. Thus, we only need to consider the height-averaged concentration defined as ∫ 1 h c dz , (2) C(x, y, t) ≡ h 0 which only depends on the lateral coordinates x and y.
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The flow of thin, non-volatile liquid films is governed by the so-called lubrication equation [103] ∂h ∂Qx ∂Qy + + =0, ∂t ∂x ∂y
(3)
where Qx ≡
h 2 τx h3 ∂p − 2µ 3µ ∂x
and
Qy ≡
h2 τy h3 ∂p − 2µ 3µ ∂y
(4)
are the volumetric flowrates, τx ≡
∂γ ∂γ ∂C = ∂x ∂C ∂x
and
τy ≡
∂γ ∂γ ∂C = ∂y ∂C ∂y
(5)
are the Marangoni stresses [86] along the x- and ydirections and ( 2 ) ∂ h ∂2h p = −γ + 2 + ρgh (6) ∂x2 ∂y is the augmented pressure, representing capillary and hydrostatic pressure contributions. Here, g = 9.81 m/s2 is the gravitational acceleration. The terms containing τx and τy represent Marangoni fluxes in response to surface tension gradients, which are caused by gradients in concentration C of the photoproducts owing to Eq. (1). Generally, the direction of Marangoni flow in thin liquid films is from regions of lower towards regions of higher surface tension. The non-uniform species distribution that sets up the surface tension gradients, however, changes continuously due to diffusion and convection. Therefore, Eq. (3) is coupled to a convection-diffusion-reaction equation [104–106] that governs the dynamics of C(x, y, t) h
∂C ∂C ∂C + Qx + Qy = Rreact (x, y, t) + ∂t ∂x ∂y ( ) ( ) ∂ ∂C ∂ ∂C + hD + hD . ∂x ∂x ∂y ∂y
Bunsen-Roscoe Law of Reciprocity, which holds for many systems [107–112]. By approximation, the reactant and photoproduct concentrations do not enter into Rreact , because at low conversion the former essentially remains constant and the latter remains very small. The parameters jbulk and Jsurf contain the intrinsic rate constant of the photochemical reaction. We note that these expressions for Rreact with constant parameters jbulk and Jsurf are valid only for conversions far below 100 %, which is appropriate for our experiments. The shape function I(x, y, t) of the UV intensity distribution depends on the beam profile as well as the aperture shape. We assume that it is time-independent during the illumination period 0 ≤ t ≤ tUV and zero afterwards. This means that in the model we neglect the presence of the chopper in the illumination system. Given the effective illumination frequency is 4 Hz and the time scale of typical experiments by far exceeds 1 s, such a time-averaging procedure is permissible (see supporting information). We approximated the shape function as I(x, y, t ≥ 0) = Θ(tUV − t, ∆t)× × Θ(|x| − 21 wx , 21 ∆w) Θ(|y| − 12 wy , 21 ∆w) ,
where Θ denotes a smoothed Heaviside function [113], wx and wy are the aperture widths in the x and y-directions and ∆t = 1 ms. We solved the set of Eqs. (3-7) numerically using the finite element software Comsol 3.5a. The dimensions of the computational domain (0 ≤ x ≤ Lx , 0 ≤ y ≤ Ly ) are typically 10 times the aperture width. The boundary conditions were chosen as ∂h =0, ∂x
(7)
The term Rreact (x, y, t) is the height-integrated photochemical conversion rate with units of length × concentration/time. We distinguish two different cases: surfacedominated and bulk-dominated reactions. In the case of a bulk-dominated reaction, the conversion occurs essentially uniformly across the film thickness. We set Rreact = hjbulk I(x, y, t), where the factor h stems from the height integration. Here, jbulk is a constant and I(x, y, t) is the scaled UV intensity distribution on the substrate. This implicitly requires the film to be optically thin (αh0 ≪ 1). In the case of a surface-dominated reaction, the conversion occurs only in a thin layer adjacent to the liquid-gas interface. The integration of the reaction rate across the film thickness is therefore independent of h and we set Rreact = Jsurf I(x, y, t), where Jsurf is assumed constant. This includes the cases of optically thick films (αh0 ≫ 1) and reactions that are truly interface-driven. In both cases we assume the reaction rate to be proportional to the light intensity, which is known as the
(8)
∂p =0 ∂x
and
∂C =0 ∂x
(9)
at x = 0 and x = Lx , and ∂h =0, ∂y
∂p =0 ∂y
and
∂C =0 ∂y
(10)
at y = 0 and y = Ly , all of which represent mirror symmetries. The boundaries at x = Lx and y = Ly are sufficiently remote from the aperture, such that the surface deformation does not reach it within the typical duration of an experiment. The initial conditions are a film of uniform thickness h(x, y, t = 0) = h0 and a uniform photoproduct concentration C(x, y, t = 0) = 0.
IV.
RESULTS AND DISCUSSION
Figure 2(a) shows a typical intensity profile I(x) corresponding to an infinitely long and 1 mm wide aperture, i.e. for wy → ∞) and wx = 1 mm. Figure 2(b) shows the corresponding film thickness profiles at different times after commencing deep-UV irradiation. The initially flat
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30
20 s 20
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N2 air
0 0
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FIG. 3. Total film thickness modulation ∆htot as a function of time for h0 = 3.6 µm and three different values of tUV = 10, 20, and 40 s. Filled circles correspond to experimental data obtained in a nitrogen atmosphere, the solid lines to two-dimensional numerical simulations using D = 3 · 10−11 m2 /s, wx = 0.9 mm, wy = 10 mm, ∆w = 0.28 mm, and Jsurf (∂γ/∂C) = 3.6 · 10−12 N/s. Open diamonds correspond to experimental data obtained in an air atmosphere for tUV = 60 s, the dotted line to two-dimensional numerical simulations using D = 3 · 10−11 m2 /s, wx = 0.9 mm, wy = 10 mm, ∆w = 0.28 mm, and Jsurf (∂γ/∂C) = 1.2 · 10−13 N/s.
103
4
Film thickness modulation ∆htot /∆Hint
1
I (x)
Film thickness h (µm)
(b)
∆hmax (nm)
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200
400
103
600
104
FIG. 2. (a) Intensity distribution I(x) for wx = 1 mm and ∆w = 0.2 mm. (b) Numerical simulations of film thickness profiles h(x, t) at different times t = 0, 20, 50, 80, 120, and 325 s for parameters h0 = 3 µm, wx = 1 mm, wy = ∞, ∆w = 0.2 mm, tUV = 60 s, D = 5 · 10−11 m2 /s, jbulk (∂γ/∂C) = 6 · 10−7 N/(m s). The violet-shaded region denotes the irradiated area. (c,d) Numerical simulations of the maximum film thickness increase ∆hmax (t) ≡ max[h] − h0 as a function of time for the same parameters as in (b). Panels (c,d) contain the same data in logarithmic and linear representations, respectively.
film develops two dimples just outside and two local maxima just inside the illuminated region. As time progresses, the dimples deepen and the maxima grow. At t ≈ 80 s the two maxima coalesce into a single maximum located at x = 0. Figure 2(b) also contains the definitions of various parameters that quantify the film thickness modulation: the maximum film thickness increase ∆hmax ≡ max[h] − h0 , the maximum film thickness reduction ∆hmin ≡ h0 − min[h], the total film thickness modulation ∆htot ≡ |∆hmax | + |∆hmin | as well as the peak-to-peak film thickness modulation ∆hpp . Figure 2(c) shows numerical simulations of ∆hmax (t) as a function of time. For t < tUV , we observe to good approximation a power-law time-dependence ∆hmax ∼ t1.57 (dashed line). The coalescence of the two maxima at t ≈ 80 s manifests itself in a steeper increase of ∆hmax .
At later times, the rate of increase of ∆hmax slows down, a peak is reached, after which ∆hmax slowly decreases. This is primarily due to lateral diffusion, which eventually leads to the disappearance of the concentration gradients that drive Marangoni flow. In Fig. 3 we present experimental data for the dependence of the total film thickness modulation ∆htot on the exposure time tUV . The film thickness modulation ∆htot was evaluated between the central maximum and one of the minima indicated in Fig. 1(b). A higher value of tUV generally leads to a higher concentration of reaction products and hence to a larger film deformation. The experimental data for tUV = 10 and 20 s are generally very well reproduced by the two-dimensional model calculations, those for 40 s are slightly overestimated. There is a striking difference between experiments performed in a dry nitrogen environment (filled circles in Fig. 3) or in an air atmosphere (open diamonds), the latter case resulting in a much smaller thickness modulation. We have repeated the experiments also in an Ar atmosphere and found that the thickness modulation is comparable to the experiments in air. We have also done experiments with the liquid film facing down, i.e. where the UV light is transmitted through the sapphire substrate before it interacts with squalane. The observed thickness modulation was only slightly reduced by about 20%. This small difference is most likely due to absorption losses in the substrate and increased reflection losses at its two surfaces. In Fig. 4 we present experimental data for the separa-
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h 0 = 4.8
= 10 s
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tUV = 10 s surface-dominated reaction model surface-dominated reaction model 200
Time (s)
400
0 0
200
4
600
400
Time [s]
(b)
FIG. 4. Separation of the film thickness minima ∆xmin as a function of time for three different values of h0 = 2.3, 3.6 and 4.8 µm and of tUV either 10, 20 or 40 s. Symbols correspond to experimental data obtained in a nitrogen atmosphere, the solid lines to two-dimensional numerical simulations using the surface-dominated reaction model and parameter values D = 3 · 10−11 m2 /s, wx = 0.9 mm, wy = 10 mm, ∆w = 0.28 mm, and Jsurf (∂γ/∂C) = 3.6 · 10−12 N/s.
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130 s
1000 s
80 s
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∆htot (nm)
0 0
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40 s t = 20 s
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tion of the film thickness minima ∆xmin as a function of time. For larger values of h0 and shorter values of tUV , ∆xmin tends to increase in time, whereas it decreases for thin films and longer irradiation times. The reason for the narrowing in the latter case is that Marangoni stresses are then strong enough to further contract the film thickness maximum against opposing capillary pressure gradients. For sufficiently long times, lateral diffusion removes concentration gradients and ∆xmin increases due to capillary pressure relaxation. The solid lines correspond to numerical 2D simulations using the surface-dominated reaction model, which reproduce the experimental data quite well. Experimental data for the dependence of the deformation amplitude on the initial film thickness are depicted in Fig. 5(a). Smaller values of h0 give rise to larger values of ∆hpp . Numerical data for the film thickness modulation as a function of h0 evaluated at fixed times are plotted in Fig. 5(b,c) based on the surface- and bulk-dominated reaction models, respectively. Two different effects are at play. On the one hand, a higher film thickness translates into a higher mobility of the liquid and thus less resistance to deformation, thereby promoting an increase in modulation. Moreover, for bulk-dominated reactions, a higher film thickness implies a larger quantity of reaction products in the irradiated region. Thus, we expect that a larger value of h0 thus tends to amplify the deformation amplitude. This expectation is confirmed in Fig. 5(c), where ∆hmax increases monotonically with h0 . The ’wiggles’ are due to the coalescence phenomenon discussed in the context of Fig. 2(b,c), which occurs at later times for thinner films. The dotted line in Fig. 5(c) corresponds to
10
(c)
3
∆hmax (nm)
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10
1000 s
300
130
80 s 40 s t = 20 s
2
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0.5
1
2
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Initial film thickness h0 (µm)
FIG. 5. (a) Peak-to-peak film thickness modulation ∆hpp as a function of time for tUV = 10 s and three different values of h0 = 2.3, 3.6 and 4.8 µm. Symbols represent experimental data obtained in a nitrogen atmosphere, solid lines 2D numerical simulations using the same parameters as in Fig. 4. (b,c) One-dimensional simulations of the film thickness modulation as a function of h0 for surface- (b) and bulk-dominated (c) reactions. The parameters that were not varied in (b) were wx = 1 mm, wy = ∞, ∆w = 0.4 mm, tUV = 60 s, D = 3 · 10−11 m2 /s, and Jsurf (∂γ/∂C) = 1.8 · 10−12 N/s. The parameters that were not varied in (c) were wx = 1 mm, wy = ∞, ∆w = 0.2 mm, tUV = 60 s, D = 5 · 10−11 m2 /s, and jbulk (∂γ/∂C) = 6 · 10−7 N/(m s). 2/3
a powerlaw ∆hmax ∼ h0 . On the other hand, in the context of the surfacedominated model, the photochemical reaction only occurs in a surface layer much thinner than h0 and the
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The solid lines in Fig. 5(a) corresponding to 2D simulations according to the surface-dominated reaction model, reproduce the experimental data well. We conclude that the ‘surface-dominated’ model is relevant to our experimental system of squalane irradiated by a deuterium lamp. This could mean that the penetration depth of the photoactive wavelengths is much smaller than h0 or that the reaction is governed by processes primarily occurring at the interface, after which the reaction products diffuse into the bulk. Figures 6(a-c) illustrate the sensitivity of the main experimental observable ∆hmax (t) with respect to variations of three key parameters: the photochemically induced rate of change of surface tension jbulk (∂γ/∂C), the diffusion coefficient D and the aperture width wx . A larger surface tension increase will lead to a higher Marangoni flow and thus a higher film thickness increase. The dashed line in Fig. 6(a) corresponds to a proportionality relation ∆hmax ∼ jbulk (∂γ/∂C) whereas the dotted line corresponds to the powerlaw ∆hmax ∼ [jbulk (∂γ/∂C)]1/3 . A larger diffusion coefficient leads to a faster broadening of the concentration profile, which reduces Marangoni stress and tends to decrease ∆hmax . The effect of D on ∆hmax depicted in Fig. 6(b) is relatively weak especially for short times t . 100 s and small values of D . 5 · 10−11 m2 /s. The diffusion coefficient becomes more important for times t comparable the diffusive timescale wx2 /D ≈ 2 · 104 s. A larger value of wx leads to a longer coalescence time tcoa of the initially separate film thickness maxima [about 80 s in Fig. 2(b)]. This implies that for a fixed instance of time t ≪ tcoa , the local height profile near the perimeter of the aperture essentially corresponds to that of an infinitely wide aperture. Thus, ∆hmax becomes independent of wx in Fig. 6(c) for sufficiently large wx . For very small values of wx , the overall quantity of photogenerated species decreases, leading to a decrease of ∆hmax with decreasing wx . The dash-dotted line in Fig. 6(c) 1/2 corresponds to a powerlaw relation ∆hmax ∼ wx . There is little qualitative difference between the two reaction models as far as the dependencies on ∂γ/∂c, D and wx are concerned. Data analogous to the ones presented in Figs. 6(a-c), but obtained with the surfacedominated model are presented as part of the supporting information.
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liquid underneath it only serves to dilute the reaction products. In this case, we expect a higher value of h0 to reduce the effective concentration and contribute to a decrease of ∆hmax . This behavior is observed in Fig. 5(b) for initial film thicknesses above approximately 3 µm. For h0 . 3 µm, the Marangoni stresses are so strong that the film almost thins completely, i.e. |∆hmin | approaches its maximum possible value h0 . The dash-dotted line in 1/2 Fig. 5(b) corresponds to a powerlaw ∆htot ∼ h0 .
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(b) 1000 s Ds 130 s
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Film thickness increase ∆hmax [nm]
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t = 20 s 10−9
Diffusion coefficient D [m2/s]
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(c)
bulk-dominated reaction model
1000 s 300 s 130 s 80 s
3
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40 s t = 20 s 2
10 0.1
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Aperture width wx [mm]
FIG. 6. (a) Numerical simulations of ∆hmax as a function of the rate of increase of surface tension jbulk (∂γ/∂C) at different times t = 20, 40, 80, 130, 300 and 1000 s. (b) ∆hmax as a function of the diffusion coefficient D at different times t = 20, 40, 80, 130 and 1000 s. The dashed vertical line indicates the self-diffusion coefficient of squalane. (c) ∆hmax as a function of aperture width wx at different times t = 20, 40, 80, 130, 300 and 1000 s. The parameters that were not varied in (a,b,c) were h0 = 3 µm, wx = 1 mm, wy = ∞, ∆w = 0.2 mm, tUV = 60 s, D = 5 · 10−11 m2 /s, jbulk (∂γ/∂C) = 6 · 10−7 N/(m s). The bulk-dominated reaction model was used.
A.
Self-similar behavior
In order to elucidate the dominant mechanisms that govern different stages of the redistribution process, we
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I(x, y, t ≥ 0) = Θ(x, ∆w) Θ(tUV − t, ∆t)
(11)
holds and first consider shallow transitions, i.e. ‘large’ values of ∆w. In this case, the concentration gradients are small and species redistribution due to both Marangoni convection and diffusion in Eq. (7) can be neglected compared to the photochemical production. This is the limit of large Damk¨ohler numbers. Thus, for the bulk-dominated reaction model, one expects C to increase linearly in time with a rate equal to jbulk C(x, t ≤ tUV ) ≈ jbulk t I(x, t ≤ tUV ) = jbulk t Θ(x, ∆w) . (12) Figure 7(a) compares the prediction of Eq. (12) with full numerical solutions of C ′ ≡ C/(jbulk t) as a function of x/∆w for 1·10−3 ≤ t ≤ 4·101 s, i.e. more than four orders of magnitude in time. The curves indeed all collapse, indicating that Eq. (12) is an excellent approximation. Consequently, the surface tension gradient can be written in the form ∂γ ∂γ ∂C ∂γ ∂Θ = ≈ tf (x) with f ≡ jbulk . (13) ∂x ∂C ∂x ∂C ∂x For large values of ∆w, flow due to capillary pressure gradients can be neglected compared to Marangoni flow in the illuminated region. Moreover, for the initially small deformation amplitudes ∆h ≡ h − h0 ≪ h0 , flow due to hydrostatic pressure gradients can also be neglected. In this case, Eq. (3) becomes separable and the Ansatz ∆h ≡ 12 t2 g(x) yields g(x) = −
h20 ∂f h2 ∂γ ∂ 2 Θ = − 0 jbulk 2µ ∂x 2µ ∂C ∂x2
(14)
in the limit ∆h ≪ h0 . This solution is represented by the black solid line in Fig. 7(b), which is an excellent approximation to the full numerical solution of the full Eqs. (3,7). For a sharp transition in the reaction rate (i.e. for ‘small’ values of ∆w), the pressure driven flow is locked on to the Marangoni flow due to the phenomenon of capillary choking [117] (see Appendix A). The Marangoni convection term scales as ( )2 ( )2 h2 ∂γ ∂C h20 ∂γ jbulk t ∂C = ∼ , (15) Qx ∂x 2µ ∂C ∂x 2µ ∂C ∆w
104 ∆hmax [nm]
∆w = 2 mm
101 (a) -0.5
1
100
0 x ∆w 0.5 /
(c) 0 -4
= ∆w
2 µm
1
(b)
10 Time [s] 1000
0.2 ∆w = 2 µm
± φ [-]
0
1/2
~t
~ 3 t /2 ∆h ~ ∆w t 2 =2 mm
1 C/( jbulk t)
derived self-similar solutions of the governing Eqs. (3,7). A finite value of the aperture width wx constitutes an imposed lengthscale, which leads to the coalescence phenomenon discussed in the context of Fig. 2(b) and which precludes self-similar behavior. Therefore, we consider here the local dynamics at the perimeter of very wide (i.e. semi-infinite) apertures. Although we can eliminate wx from the problem in this fashion, we do have the other (imposed) lengthscale ∆w to consider. Consequently, we can only identify self-similar solutions that are restricted in their validity to certain time intervals, when the solution is either no longer or not yet affected by the value of ∆w. We assume that locally
C’ [-]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
-φ off
φ on
0
0
ξ [-]
4
-0.2 −8
(d) 0
η [-]
8
FIG. 7. (a) Comparison of Eq. (12) (red solid line) with corresponding numerical solutions of the full non-linear Eqs. (3,7) (blue symbols) for times 10−3 ≤ t ≤ 40 s. For the latter, the relevant parameter values were h0 = 3 µm, ∆w = 2 mm, tUV = 600 s, D = 5 · 10−11 m2 /s, jbulk (∂γ/∂C) = 6 · 10−7 N/(m s). (b) Film thickness modulation ∆hmax (t) for two examples of a very shallow (∆w = 2 mm, tUV = 600 s, D = 5 · 10−11 m2 /s) and a very abrupt (∆w = 2 µm, tUV = 10 s, D = 5 · 10−13 m2 /s) transition. The other parameter values were h0 = 3 µm, jbulk (∂γ/∂C) = 6 · 10−7 N/(m s). (c) Self-similar solution of Eq. (17) (solid line) as well as corresponding numerical solutions (symbols) of the full non-linear Eqs. (3,7) for times 0.01 ≤ t ≤ 10 s. For the latter, the relevant parameter values were h0 = 3 µm, ∆w = 2 µm, tUV = 60 s, D = 5 · 10−11 m2 /s, jbulk (∂γ/∂C) = 6 · 10−7 N/(m s). (d) Self-similar solutions ϕon (κ = 1, solid line) and −ϕoff (κ = 0, dashed line) of Eq. (A5) as well as corresponding numerical solutions (symbols) of the full non-linear Eqs. (3,7) for times 10−4 ≤ t ≤ 10 s (green circles) and 40 ≤ t ≤ 1000 s (red squares), respectively. For the latter, the relevant parameter values were h0 = 3 µm, ∆w = 2 µm, tUV = 10 s, D = 5 · 10−13 m2 /s, jbulk (∂γ/∂C) = 6 · 10−7 N/(m s).
i.e. quadratically with time t. Consequently, both convective terms are negligible compared to the diffusive jbulk t term scaling as D (∆w) 2 (i.e. linearly with t) at early times. This corresponds to the limit of a large convective Damk¨ohler number and a diffusive Damk¨ohler number of order 1. Thus, in the limit of ∆h ≪ h0 or equivalently h ≈ h0 , the 1D reaction-convection-diffusion equation can be simplified to ∂C ∂2C = jbulk I(x) + D 2 . ∂t ∂x
(16)
For an abrupt transition, the intensity distribution I(x) and thus the reaction rate term are scale-invariant, i.e. I(x) = I(kx), for arbitrary positive numbers k. InC troducing the reduced concentration C ′ ≡ jbulk t allows us to derive (see Appendix A) the following ordinary differential equation (ODE) in the self-similar coordinate
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ξ ≡ x/(Dt)1/2 −
d2 C ′ ξ dC ′ + C ′ = I(ξ) + . 2 dξ dξ 2
(17)
The corresponding self-similar solution for C ′ (ξ) is plotted as the solid line in Fig. 7(c) alongside with numerical solutions of the full set of Eqs. (3,7) for three decades in time 0.01 ≤ t ≤ 10 s (symbols). Next we determine a similarity solution for the evolution of the height profile h(x, t). A small value of ∆w implies that initially the Marangoni stress driving the film deformation is present only in an exceedingly narrow interval, outside of which the only driving force is given by capillary pressure gradients. Consequently, at the borders of this interval, a flux continuity condition must hold, which provides the above-mentioned linkage between capillary and Marangoni fluxes [117]. If we set τx ≡ Atκ δ(x) and seek a self-similar solution in the variable ϕ(η) ≡ ∆h/(Dβ tβ ), where η ≡ Etxα is the self-similarity coordinate, then its validity in the region outside the narrow interval implies the capillary scaling α = 1/4 [114–116]. Here, Dβ and E are constants that render ϕ and η dimensionless. The flux continuity determines the value of β = κ + 12 . For details, we refer the reader to Appendix A. The self-similar solution for C ′ ∼ t C(ξ) is decoupled from ∆h(x, t), because at early times convective mass fluxes can be neglected compared to the reactive and diffusive contributions. Consequently, the effective concentration difference between illuminated and unilluminated regions scales as ∆C (t ≤ tUV ) ∼ t1 or ∆C (t > tUV ) = const ∼ t0 before and after switching off the UV light, respectively. If the width of the transition zone of the concentration distribution can be considered an exceedingly narrow interval despite its diffusive broadening (as far as the flow field is concerned), then the exponents in the relation τx = Atκ δ(x) have values of κon = 1 and κoff = 0. This implies βon = 3/2 and βoff = 1/2, respectively. The dashed and dotted lines Fig. 7(b) correspond to powerlaw relations ∆h ∼ t3/2 and ∆h ∼ t1/2 , respectively, which reproduce the behavior of the full numerical solutions (symbols) very well. Also the time exponent of 1.57 in Fig. 2(c) is close to 3/2, the origin of the deviation is that ∆w = 0.2 mm is not sufficiently abrupt. We have also compared the full numerical solutions of Eqs. (3,7) for ∆w = 2 µm depicted in Fig. 7(b) with the self-similar height profiles ϕ(η) determined from solving the Eq. (A5). The solid line in Fig. 7(d) represents ϕon , i.e. the on-stage (β = 3/2), which is an excellent approximation to the numerical data (symbols) for 5 decades in time 10−4 ≤ t ≤ 10 s. The dashed line represents −ϕoff , i.e. the off-stage (β = 1/2), which approximates the amplitude and shape of the numerical data (symbols) very well for 40 ≤ t ≤ 1000 s. We inverted the sign of ϕoff merely to avoid excessive overlap between the curves. The apparent lateral shift between −ϕoff and the numerical data is caused by the convection of the concentration distribution, which Eq. (A5) does not account for.
Finally we note that in the limit of small deformations, the self-similar solutions derived above are equally relevant to the surface- and the bulk-dominated reaction models, since the structure of Eq. (7) is identical in the limit h ≈ h0 .
B.
Reaction mechanism
Deep UV irradiation in an air environment can lead to ozone formation and partial oxidation of hydrocarbon surfaces as exploited technologically for surface cleaning.[1] Thus, our initial expectation was that ozone generation and subsequent oxidation of squalane is a likely candidate for the dominant reaction pathway. In striking contrast, the experimental results obtained in air and nitrogen atmospheres shown in Fig. 3 indicate that oxygen plays more the role of an inhibitor rather than promotor of the reaction. Consistent with our findings in Fig. 3, Bruggeman et al. state that the presence of oxygen may become a rate limiting factor due to its reactivity with short-lived hydrogen and carbon radicals [125]. Generally, photochemical reactions are complex and require sophisticated experimental instrumentation for the elucidation of reaction mechanisms, rates and product distributions [118–125]. Yang et al. studied the photolysis of liquid cyclohexane at 147 nm, the primary product being cyclohexene. They concluded that the presence of benzene caused a drastic reduction in the rate of cyclohexane decomposition due to scavenging of hydrogen atoms [118]. Holroyd determined the principal primary process at 147 nm in n-pentane to be H2 elimination and the formation of hydrogen atoms and pentyl radicals [119]. Nurmukhametov et al. exposed polystyrene films and solutions to 248 nm radiation and concluded that the UV-induced reaction includes dehydrogenation of the molecular chain and the subsequent formation of polyconjugated polyene chains [123]. Bossa et al. developed and validated a kinetic model for methane ice photochemistry upon irradiation with UV light in the 120 - 200 nm wavelength range [124]. We therefore suspect that alkene formation might occur in our experiments. Consistent with this hypothesis, Birdi found that the surface tension values of n-alkenes are systematically higher than those of corresponding n-alkanes by approximately 0.5 - 1% [126]. One possible conclusion from Fig. 5(a) was that squalane is an optically thick material for deep UV radiation. Moreover, we mentioned that the measured thickness modulation is comparable for the liquid film facing up or facing down. This may suggest that the crucial requirement for the reaction is not so much the presence or absence of ambient gases, but rather the presence or absence of dissolved gases in the liquid. Given the small liquid film thickness, the diffusion of gases into squalane after spin-coating only requires less than one second, which is much shorter than the time required for
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mounting the sample.
C.
V.
Implications for reaction monitoring
The experimentally adjustable parameters of this technique are the initial film thickness h0 , the aperture size wx , and the illumination time tUV . The extractable parameters are the surface tension change ∆γ = γ − γ0 and the diffusion coefficient D. Our experiments were restricted to a constant aperture width wx ≈ 1 mm, which implied that they had a low sensitivity towards variations of D. According to our simulations, however, which are well validated by the gathered experimental data, we expect that smaller slit widths allow more conclusive information regarding the value of D. The achievable time resolution is determined by how quickly a measurable grayscale modulation is observed in the interferometry images. Based on the numerical results and a required minimum thickness modulation ∆htot ≈ 50 nm, we estimate that a time resolution of several seconds is feasible for the relatively weak surface tension changes occurring in squalane. We have specifically selected squalane in order to demonstrate that relatively subtle chemical composition changes can be detected conveniently. According to Eq. (14), the response time will be faster for more intense irradiation (e.g. by using laser sources), for reactions that exhibit higher conversion rates or more surface-active reaction products, for thicker films, lower viscosities and smaller values of ∆w. We note that the oblong geometry of the aperture (wx = 1 mm ≪ wy = 10 mm < ∞) enhances the sensitivity of the method compared to a 1D geometry (wy → ∞) by up to approximately 30% for wx ≈ 1 mm, as the end-effect increases the height of the maximum. The maximum is free to stretch along the y-direction, thereby minimizing reductions of ∆hmax induced by capillary confinement. By comparison, for a square aperture (wx = wy = 1 mm), the enhancement compared to a 1D geometry is only a few percent for typical parameter values. We note further that in the dilute regime considered in this manuscript, the method is only sensitive to the products jbulk (∂γ/∂C) and Jsurf (∂γ/∂C), but not independently to e.g. Jsurf and ∂γ/∂C. This is a consequence of the fact that the concentration of the reaction product scales with the production rate constants jbulk and Jsurf . The lubrication equation is coupled to the concen∂γ ∂C tration field via the Marangoni stresses, e.g. τx = ∂C ∂x , which thus only depend on the above-mentioned products. Consequently, the height profile h and the fluxes Qx and Qy , which enter into Eq. (7), only depend on the products jbulk (∂γ/∂C) and Jsurf (∂γ/∂C). The same, therefore, holds for the Peclet number inherent in Eq. (7).
SUMMARY
We have studied broadband deep-UV irradiation of thin liquid films of the aliphatic hydrocarbon squalane through a slit aperture. This served as a model system for monitoring photochemical reactions based on interferometric detection of film thickness deformations. The mechanism driving material redistribution is solutocapillary Marangoni flow as a consequence of surface tension changes induced by the reaction products. Due to the differential nature of Marangoni flows, a high sensitivity of the technique can be achieved. Surface tension changes as small as ∆γ = 10−6 N/m can be detected and time-resolutions below one second can be achieved. We performed systematic experiments where we varied the film thickness and illumination time. Interestingly, experiments performed in a nitrogen atmosphere resulted in a significantly stronger film deformation than experiments performed in either air or Ar environments. We also developed a model of the photochemical reaction and the convection and diffusion of the reaction products. Although the model takes into account only a single photoproduct species, it reproduces the experimental results almost quantitatively, if we assume that the reaction occurs primarily in a region close to the liquid surface that is much smaller than the film thickness. The main output parameter of the numerical simulations is the product of the reaction rate and the rate of change of surface tension with the photoproduct concentration. By using narrower aperture widths, also information on the diffusion coefficient of the photoproduct species could be gathered. Besides performing numerical simulations, we also derived and numerically validated self-similar solutions of the governing equations for the concentration and the height profiles, which characterize the early-time response.
ACKNOWLEDGMENTS
The authors gratefully acknowledge that this research is supported partially by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs. The authors thank Jϕrgen van der Veen and Henny Manders for their help with constructing the experimental setup.
SUPPORTING INFORMATION
Video of a typical experiment. Data supporting the negligible difference between intermittent and continuous illumination. Further simulation results in analogy to Fig. 6 but using the surface-dominated reaction model. Supporting information is available free of charge via the Internet at http://pubs.acs.org/.
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Appendix A: Self-similar solutions for abrupt transitions
Introducing the dimensionless concentration C ′ = jbulk t implies C
lubrication equation can be linearized and is given by ( ) ∂∆h ∂ h20 τ h30 γ ∂ 3 ∆h + + =0. (A2) ∂t ∂x 2µ 3µ ∂x3 We seek a solution of the form ∆h = Dβ tβ ϕ(η) ,
′
and
∂C 1 ∂C C = − or equivalently ∂t jbulk t ∂t jbulk t2 ∂C ∂C ′ C ∂C ′ = jbulk t + = jbulk t + jbulk C ′ ∂t ∂t t ∂t ∂2C ∂2C ′ D 2 = Djbulk t . ∂x ∂x2
Equation (16) can be converted into an ODE by introducing the self-similar coordinate ξ ≡ x/(Dt)1/2 , leading to [ ] [ ] ∂C ′ ∂C ′ ∂ξ ∂C ′ Dx ∂C ′ ξ = = − = − ∂t ∂ξ ∂t ∂ξ ∂ξ 2t 2(Dt)3/2 ′ ′ ′ 1 ∂C ∂ξ ∂C ∂C = = , ∂x ∂ξ ∂x ∂ξ (Dt)1/2 ∂2C ′ 1 ∂2C ′ = 2 ∂x Dt ∂ξ 2 and thus (
)
dC ′ d2 C ′ 1 + jbulk C ′ = jbulk I(ξ) + jbulk t 2 dξ dξ t 2 ′ ξ dC ′ d C =⇒ − + C ′ = I(ξ) + . (A1) 2 dξ dξ 2
ξ jbulk t − 2t
The lubrication equation for a narrow peak in the Marangoni stress of the form τ = Atκ δ(x) allows for a self-similar solution as well. Here, δ(x) represents the Dirac delta function and A is a constant the dimension of which depends on the value of the exponent κ. We consider a thin liquid film of initially uniform thickness h0 and small deformations ∆h ≪ h0 . In this case the
[1] J. R. Vig. UV/ozone cleaning of surfaces. J. Vac. Sci. Technol. A 1985, 3, 1027-1034. [2] Kim, C.-K.; Chung, C.-H.; Moon, S. H. Removal of Organic Impurities from the Silicon Surface by Oxygen and UV Cleaning. Korean J. Chem. Eng. 1996, 13, 328-330. [3] Fominski, V. Yu.; Naoumenko, O. I.; Nevolin, V. N.; Alekhin, A. P.; Markeev, A. M.; Vyukov, L. A. Photochemical removal of organic contaminants from silicon surface at room temperature. Appl. Phys. Lett. 1996, 68, 2243-2245.
(A3)
where η ≡ Etxα and Dβ and E are constants that render ϕ and η dimensionless. Its validity in the region where the Marangoni stress is zero [114–116] implies α = 1/4. At the boundary of the narrow region, the flux must be continuous, which implies that h3 γ ∂ 3 ∆h h20 τ ∼ 0 2µ 3µ ∂x3
=⇒
tκ−α ∼ tβ−3α
(A4)
or β = κ + 2α. Let us now transform Eq. (A2) into a corresponding ODE in the self-similar coordinate η. We have ] ∂ [ dϕ ∂η Dβ tβ ϕ(η) = Dβ βtβ−1 ϕ + Dβ tβ ∂t dη ∂t [ ] dϕ = Dβ tβ−1 βϕ − αη dη ] dϕ ∂η D ∂ [ β β−α dϕ Dβ tβ ϕ(η) = Dβ tβ = t ∂x dη ∂x E dη ( )4 4 [ 4 ] ∂ d ϕ ∂η Dβ d4 ϕ Dβ tβ ϕ(η) = Dβ tβ 4 = 4 tβ−4α 4 4 ∂x dη ∂x E dη κ−2α ∂ At d ∂τ = [Atκ δ(x)] = [δ(η)] , ∂x ∂x E 2 dη where we have used δ(kx) = δ(x)/|k|. The ODE thus becomes [ ] dϕ d h20 A 1 h30 γ d3 ϕ βϕ − αη + δ(η) + 4 =0 dη dη 2µDβ E 2 E 3µ dη 3 √ Setting E ≡ 4 h30 γ/(3µ) and Dβ ≡ h20 A/(2µE 2 ) = √ √ A 3h0 / 4µγ yields [ ] dϕ d d3 ϕ βϕ − αη + δ(η) + 3 = 0 . (A5) dη dη dη
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