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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 5600MB, The Netherlands S Supporting Information *

ABSTRACT: We evaluated the sensitivity and time resolution of a technique for photochemical reaction monitoring based on the interferometric detection of the deformation of liquid films. The reaction products change the local surface tension and induce 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 Δγ = 10−6 N/m can be detected, and time resolutions of 0. This is typically a realistic assumption for nonaqueous 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 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 time scale of the experiments typically ranges from 10 to 600 s, which by far exceeds the diffusive time scale h0 2 /Ds ≲ 0.8s. Consequently, in both cases the concentration distribution can be assumed to be vertically equilibrated. Thus, we need to consider only the height-averaged concentration defined as

thermal conductivity of k ≈ 25 W/(mK), which helps to 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 tension99 is γ = 28.15 mN/m, the density100 is ρ = 805 kg/m3, and the refractive index is nD = 1.452. The self-diffusion coefficient of squalane101 is Ds ≈ 3 × 10−11 m2/s. Kowert and Watson studied the diffusion of organic solutes in squalane.102 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 the absorption of deep-UV radiation. The effective emission wavelengths range approximately from 160 to 400 nm. The lamp output does not exhibit any significant start-up effects for illumination times of less than 1 min. 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 in either 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 path length 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 exposure, 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. Because 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 1b shows an example of an interference image. The height difference represented by two consecutive constructive or destructive interference fringes corresponds to ΔHint = 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 Figure 1b and multiplication by ΔHint.

C(x , y , t ) ≡

1 h

∫0

h

c dz

(2)

which depends only on the lateral coordinates x and y. The flow of thin, nonvolatile liquid films is governed by the so-called lubrication equation103 ∂Q y ∂Q x ∂h =0 + + ∂y ∂t ∂x

(3)

where Qx ≡

h2τx h3 ∂p − 2μ 3μ ∂x

Qy ≡

and

h2τy 2μ

−

h3 ∂p 3μ ∂y

(4)

are the volumetric flow rates, τx ≡

∂γ ∂γ ∂C = ∂x ∂C ∂x

and

are the Marangoni stresses

86

τy ≡

∂γ ∂γ ∂C = ∂y ∂C ∂y

(5)

along the x and y directions and

⎛ ∂ 2h ∂ 2h ⎞ p = −γ ⎜ 2 + 2 ⎟ + ρgh ∂y ⎠ ⎝ ∂x

(6)

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 toward regions of higher surface tension. The nonuniform 3648

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∂p ∂h = 0, =0 ∂x ∂x

species distribution that sets up the surface tension gradients, however, changes continuously because of diffusion and convection. Therefore, eq 3 is coupled to a convection− diffusion-reaction equation104−106 that governs the dynamics of C(x, y, t) h

∂ ⎛⎜ ∂C ⎞⎟ ∂ ⎛ ∂C ⎞ + hD ⎜h D ⎟ ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠

∂C =0 ∂x

(9)

at x = 0 and x = Lx and ∂p ∂h = 0, = 0, ∂y ∂y

∂C ∂C ∂C + Qx + Qy = R react(x , y , t )+ ∂t ∂x ∂y +

and

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.

(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: surface-dominated and bulkdominated reactions. In the case of a bulk-dominated reaction, the conversion occurs essentially uniformly across the film thickness. We set Rreact = hjbulkI(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 = JsurfI(x, y, t), where Jsurf is assumed to be 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 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 of 0 ≤ t ≤ tUV and zero afterward. This means that in the model we neglect the presence of the chopper in the illumination system. Given that 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 (Supporting Information). We approximated the shape function as

IV. RESULTS AND DISCUSSION Figure 2a 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 2b shows the corresponding film thickness profiles at different times after commencing deep-UV

I(x , y , t ≥ 0) = Θ(t UV − t , Δt ) × ⎛1 ⎞ ⎛1 ⎞ 1 1 Θ⎜ wx − |x| , Δw⎟ Θ⎜ wy − |y| , Δw⎟ ⎝2 ⎠ ⎝ ⎠ 2 2 2

Figure 2. (a) Intensity distribution I(x) for wx = 1 mm and Δw = 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 = ∞, Δw = 0.2 mm, tUV = 60 s, D = 5 × 10−11 m2/ s, and 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 Δhmax(t) ≡ max[h] − h0 as a function of time for the same parameters as in panel b. Panels c and d contain the same data in logarithmic and linear representations, respectively.

(8)

where Θ denotes a smoothed Heaviside function,113 wx and wy are the aperture widths in the x and y directions, and Δt = 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 3649

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irradiation. The initially flat 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 2b also contains the definitions of various parameters that quantify the film thickness modulation: the maximum film thickness increase Δhmax ≡ max[h] − h0, the maximum film thickness reduction Δhmin ≡ h0 − min[h], the total film thickness modulation Δhtot ≡ |Δhmax| + |Δhmin|, and the peak-to-peak film thickness modulation Δhpp. Figure 2c shows numerical simulations of Δhmax(t) as a function of time. For t < tUV, we observe to good approximation a power law time dependence of Δhmax ∼ t1.57 (dashed line). The coalescence of the two maxima at t ≈ 80 s manifests itself in a steeper increase in Δhmax. At later times, the rate of increase of Δhmax slows down and a peak is reached, after which Δhmax slowly decreases. This is primarily due to lateral diffusion, which eventually leads to the disappearance of the concentration gradients that drive Marangoni flow. In Figure 3, we present experimental data for the dependence of the total film thickness modulation Δhtot on the exposure

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 Figure 4, we present experimental data for the separation of the film thickness minima Δxmin as a function of time. For

Figure 4. Separation of the film thickness minima Δxmin as a function of time for three different values of h0 = 2.3, 3.6, and 4.8 μm and of tUV = 10, 20, or 40 s. Symbols correspond to experimental data obtained in a nitrogen atmosphere, and the solid lines correspond to twodimensional numerical simulations using the surface-dominated reaction model and parameter values D = 3 × 10−11 m2/s, wx = 0.9 mm, wy = 10 mm, Δw = 0.28 mm, and Jsurf(∂γ/∂C) = 3.6 × 10−12 N/s.

larger values of h0 and shorter values of tUV, Δxmin 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 Δxmin increases as a result of capillary pressure relaxation. The solid lines correspond to numerical 2D simulations using the surfacedominated 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 Figure 5a. Smaller values of h0 give rise to larger values of Δhpp. Numerical data for the film thickness modulation as a function of h0 evaluated at fixed times are plotted in Figure 5b,c on the basis of the surface- and bulk-dominated reaction models, respectively. Two different effects are at play. On the one hand, a greater film thickness translates to a higher mobility of the liquid and thus less resistance to deformation, thereby promoting an increase in modulation. Moreover, for bulkdominated reactions, a greater 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 Figure 5c, where Δhmax increases monotonically with h0. The “wiggles” are due to the coalescence phenomenon discussed in the context of Figure 2b,c, which occurs at later times for thinner films. The dotted line in Figure 5c corresponds to a power law of Δhmax ∼ h02/3.

Figure 3. Total film thickness modulation Δhtot 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, and the solid lines correspond to two-dimensional numerical simulations using D = 3 × 10−11 m2/s, wx = 0.9 mm, wy = 10 mm, Δw = 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, and the dotted line corresponds to twodimensional numerical simulations using D = 3 × 10−11 m2/s, wx = 0.9 mm, wy = 10 mm, Δw = 0.28 mm, and Jsurf(∂γ/∂C) = 1.2 × 10−13 N/s.

time tUV. The film thickness modulation Δhtot was evaluated between the central maximum and one of the minima indicated in Figure 1b. 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, but those for 40 s are slightly overestimated. There is a striking difference between experiments performed in a dry nitrogen environment (filled circles in Figure 3) or in an air atmosphere (open diamonds), with the latter case resulting in a much smaller thickness modulation. We have also repeated the experiments 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., 3650

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maximum possible value h0. The dashed−dotted line in Figure 5b corresponds to a power law of Δhtot ∼ h01/2. The solid lines in Figure 5a corresponding to 2D simulations according to the surface-dominated reaction model reproduce the experimental data well. We conclude that the surfacedominated 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. Figure 6a−c illustrates the sensitivity of the main experimental observable Δhmax(t) with respect to variations in 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 Figure 6a corresponds to a proportionality relation Δhmax ∼ jbulk(∂γ/∂C), whereas the dotted line corresponds to the power law Δhmax ∼ [jbulk(∂γ/ ∂C)]1/3. A larger diffusion coefficient leads to a faster broadening of the concentration profile, which reduces the Marangoni stress and tends to decrease Δhmax. The effect of D on Δhmax depicted in Figure 6b is relatively weak, especially for short times t ≲ 100 s and small values of D ≲ 5 × 10−11m2/s. The diffusion coefficient becomes more important for times t comparable to the diffusive time scale wx2/D ≈ 2 × 104s. A larger value of wx leads to a longer coalescence time tcoa of the initially separate film thickness maxima (about 80 s in Figure 2b). 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, Δhmax becomes independent of wx in Figure 6c for sufficiently large wx. For very small values of wx, the overall quantity of photogenerated species decreases, leading to a decrease in Δhmax with decreasing wx. The dashed−dotted line in Figure 6c corresponds to the power law relation Δhmax ∼ wx1/2. 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 that presented in Figure 6a−c but obtained with the surface-dominated model are presented as part of the Supporting Information. A. Self-Similar Behavior. To elucidate the dominant mechanisms that govern different stages of the redistribution process, we derived self-similar solutions of the governing equations (eqs 3 and 7). A finite value of the aperture width wx constitutes an imposed length scale, which leads to the coalescence phenomenon discussed in the context of Figure 2b and precludes self-similar behavior. Therefore, we consider here the local dynamics at the perimeter of very wide (i.e., semiinfinite) apertures. Although we can eliminate wx from the problem in this fashion, we do have the other (imposed) length scale Δw to consider. Consequently, we can identify only selfsimilar 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 Δw. We assume that locally

Figure 5. (a) Peak-to-peak film thickness modulation Δhpp as a function of time for tUV = 10 s and h0 = 2.3, 3.6, and 4.8 μm. Symbols represent experimental data obtained in a nitrogen atmosphere, and solid lines represent 2D numerical simulations using the same parameters as in Figure 4. (b, c) One-dimensional simulations of the film thickness modulation as a function of h0 for surface- (b) and bulkdominated (c) reactions. The parameters that were not varied in panel b were wx = 1 mm, wy = ∞, Δw = 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 panel c were wx = 1 mm, wy = ∞, Δw = 0.2 mm, tUV = 60 s, D = 5 × 10−11 m2/s, and jbulk(∂γ/∂C) = 6 × 10−7 N/(m s).

On the other hand, in the context of the surface-dominated model, the photochemical reaction occurs only in a surface layer much thinner than h0, and the liquid underneath it serves only 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 in Δhmax. This behavior is observed in Figure 5b for initial film thicknesses above approximately 3 μm. For h0 ≲ 3 μm, the Marangoni stresses are so strong that the film thins almost completely, i.e., |Δhmin| approaches its

I(x , y , t ≥ 0) = Θ( − x , Δw) Θ(t UV − t , Δt )

(11)

holds and first consider shallow transitions, i.e., large values of Δw. In this case, the concentration gradients are small, and 3651

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Figure 7. (a) Comparison of eq 12 (red solid line) with corresponding numerical solutions of the full nonlinear equations (eqs 3 and 7) (blue symbols) for times 10−3 ≤ t ≤ 40 s. For the latter, the relevant parameter values were h0 = 3 μm, Δw = 2 mm, tUV = 600 s, D = 5 × 10−11 m2/s, and jbulk(∂γ/∂C) = 6 × 10−7 N/(m s). (b) Film thickness modulation Δhmax(t) for two examples of a very shallow (Δw = 2 mm, tUV = 600 s, D = 5 × 10−11m2/s) and a very abrupt (Δw = 2 μm, tUV = 10 s, D = 5 × 10−13m2/s) transition. The other parameter values were h0 = 3 μm and 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 nonlinear equations (eqs 3 and 7) for times 0.01 ≤ t ≤ 10 s. For the latter, the relevant parameter values were h0 = 3 μm, Δw = 2 μm, tUV = 60 s, D = 5 × 10−11 m2/s, and 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 nonlinear equations (eqs 3 and 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, Δw = 2 μm, tUV = 10 s, D = 5 × 10−13 m2/s, and jbulk(∂γ/∂C) = 6 × 10−7 N/(m s).

The curves indeed all collapse, indicating that eq 12 is an excellent approximation. Consequently, the surface tension gradient can be written in the form

Figure 6. (a) Numerical simulations of Δhmax 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) Δhmax 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) Δhmax 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 panel a−c were h0 = 3 μm, wx = 1 mm, wy = ∞, Δw = 0.2 mm, tUV = 60 s, D = 5 × 10−11 m2/s, and jbulk(∂γ/∂C) = 6 × 10−7 N/(m s). The bulkdominated reaction model was used.

∂γ ∂γ ∂C ∂γ ∂Θ = ≈ tf (x) with f ≡ jbulk ∂x ∂C ∂x ∂C ∂x

(13)

For large values of Δw, flow due to capillary pressure gradients can be neglected compared to Marangoni flow in the illuminated region. Moreover, for the initially small deformation amplitudes Δh ≡ h − h0 ≪ h0, flow due to hydrostatic pressure gradients can also be neglected. In this case, eq 3 becomes 1 separable and the Ansatz Δh ≡ 2 t 2g (x) yields

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 Damköhler numbers. Thus, for the bulk-dominated reaction model, one expects C to increase linearly in time with a rate equal to jbulk

g (x ) = −

h0 2 ∂f h2 ∂γ ∂ 2Θ = − 0 jbulk ∂C ∂x 2 2μ ∂x 2μ

(14)

in the limit of Δh ≪ h0. This solution is represented by the black solid line in Figure 7b, which is an excellent approximation to the full numerical solution of the full equations (eqs 3 and 7). For a sharp transition in the reaction rate, i.e., for small values of Δw, the pressure-driven flow is locked onto the Marangoni flow because of the phenomenon of capillary choking117 (Appendix A). The Marangoni convection term scales as

C(x , t ≤ t UV ) ≈ jbulk tI(x , t ≤ t UV ) = jbulk t Θ( − x , Δw) (12)

Figure 7a compares the prediction of eq 12 with full numerical solutions of C′ ≡ C/(jbulkt) as a function of x/Δw for 1 × 10−3 ≤ t ≤ 4 × 101 s, i.e., more than 4 orders of magnitude in time. 3652

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2 2 h 2 ∂γ ⎛ jbulk t ⎞ h2 ∂γ ⎛⎜ ∂C ⎞⎟ ∂C ∼ 0 = ⎜ ⎟ 2μ ∂C ⎝ ∂x ⎠ 2μ ∂C ⎝ Δw ⎠ ∂x

Article

height profiles ϕ(η) determined from solving eq A5. The solid line in Figure 7d 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 progressive 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 bulk-dominated reaction models because the structure of eq 7 is identical in the limit of h ≈ h0. B. Reaction Mechanism. Deep UV irradiation in an air environment can lead to ozone formation and the partial oxidation of hydrocarbon surfaces as exploited technologically for surface cleaning.1 Thus, our initial expectation was that ozone generation and the subsequent oxidation of squalane are likely candidates for the dominant reaction pathway. In striking contrast, the experimental results obtained in air and nitrogen atmospheres shown in Figure 3 indicate that oxygen plays more the role of an inhibitor rather than a promotor of the reaction. Consistent with our findings in Figure 3, Bruggeman et al. state that the presence of oxygen may become a rate-limiting factor because of 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, with the primary product being cyclohexene. They concluded that the presence of benzene caused a drastic reduction in the rate of cyclohexane decomposition due to the 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 UVinduced reaction includes the 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 Figure 5a 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 requires less than 1 s, which is much shorter than the time required for mounting the sample. C. 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

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i.e., quadratically with time t. Consequently, both convective terms are negligible compared to the diffusive term scaling as Djbulkt/(Δw)2, i.e., linearly with t, at early times. This corresponds to the limit of a large convective Damköhler number and a diffusive Damköhler number of order 1. Thus, in the limit of Δh ≪ 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. Introducing the reduced concentration C′ ≡ C/(jbulkt) allows us to derive (Appendix A) the following ordinary differential equation (ODE) in the self-similar coordinate ξ ≡ x/(Dt)1/2 −

ξ dC ′ d2C′ + C ′ = I (ξ ) + 2 dξ dξ 2

(17)

The corresponding self-similar solution for C′(ξ) is plotted as the solid line in Figure 7c alongside numerical solutions of the full set of equations (eqs 3 and 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 Δw 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) ≡ Δh/(Dβtβ), where η ≡ x/(Etα) 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 1 continuity determines the value of β = κ + 2 . For details, we refer the reader to Appendix A. The self-similar solution for C ∼ tC′(ξ) is decoupled from Δh(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 ΔC (t ≤ tUV) ∼ t1 or ΔC (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 to be an exceedingly narrow interval despite its diffusive broadening (as far as the flow field is concerned), then the exponents in the relation τx = Atκδ(x) have values of κon = 1 and κoff = 0. This implies βon = 3/2 and βoff = 1/2, respectively. The dashed and dotted lines Figure 7b correspond to power law relations Δh ∼ t3/2 and Δh ∼ t1/2, respectively, which reproduce the behavior of the full numerical solutions (symbols) very well. Also, the time exponent of 1.57 in Figure 2c is close to 3/2, and the origin of the deviation is that Δw = 0.2 mm is not sufficiently abrupt. We have also compared the full numerical solutions of eqs 3 and 7 for Δw = 2 μm depicted in Figure 7b with the self-similar 3653

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Δγ = 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 toward variations in 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. On the basis of the numerical results and a required minimum thickness modulation Δhtot ≈ 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 Δw. 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 because the end effect increases the height of the maximum. The maximum is free to stretch along the y direction, thereby minimizing reductions of Δhmax induced by capillary confinement. By comparison, for a square aperture (wx = wy = 1 mm), the enhancement compared to the 1D geometry is only a few percent for typical parameter values. We note further that in the dilute regime considered in this article the method is sensitive only to products jbulk(∂γ/∂C) and Jsurf(∂γ/∂C) but not independently to 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 concentration field ∂γ ∂C via the Marangoni stresses, e.g. τx = ∂C ∂x , which thus depend only on the above-mentioned products. Consequently, the height profile h and fluxes Qx and Qy, which enter into eq 7, depend only on products jbulk(∂γ/∂C) and Jsurf(∂γ/∂C). The same, therefore, holds for the Peclet number inherent in eq 7.

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, information on the diffusion coefficient of the photoproduct species could also 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.

■

APPENDIX A: SELF-SIMILAR SOLUTIONS FOR ABRUPT TRANSITIONS

Introducing the dimensionless concentration C′ = ∂C′ C 1 ∂C − = ∂t jbulk t ∂t jbulk t 2

C jbulk t

implies

or equivalently

∂C ∂C′ ∂C′ C = jbulk t + = jbulk t + jbulk C′ ∂t ∂t ∂t t 2 2 ∂C ∂ C′ D 2 = Djbulk t 2 ∂x ∂x

and

Equation 16 can be converted to an ODE by introducing the self-similar coordinate ξ ≡ x/(Dt)1/2, leading to Dx ⎤ ∂C′ ∂ξ ∂C′ ⎡ ∂C′ ⎡ ξ ⎤ ∂C′ ⎥= ⎢− = = ⎢− ⎥ 3/2 ∂t ∂ξ ∂t ∂ξ ⎣ 2(Dt ) ⎦ ∂ξ ⎣ 2t ⎦ ∂C′ ∂C′ ∂ξ ∂C′ 1 = = ∂x ∂ξ ∂x ∂ξ (Dt )1/2 1 ∂ 2C′ ∂ 2C′ = Dt ∂ξ 2 ∂x 2

and thus ⎛ ξ ⎞ dC ′ d2C′ 1 + jbulk C′ = jbulk I(ξ) + jbulk t 2 jbulk t ⎜ − ⎟ ⎝ 2t ⎠ dξ dξ t ⇒−

V. 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 the 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. Because of the differential nature of Marangoni flows, a high sensitivity of the technique can be achieved. Surface tension changes of as small as Δγ = 10−6 N/m can be detected, and time resolutions below 1 s 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

ξ dC ′ d2C′ + C ′ = I (ξ ) + 2 dξ dξ 2 (A1)

The lubrication equation for a narrow peak in the Marangoni stress of the form τ = Atκδ(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 Δh ≪ h0. In this case, the lubrication equation can be linearized and is given by 2 h 3γ ∂ 3Δh ⎞ ∂Δh ∂ ⎛ h0 τ ⎜ ⎟=0 + + 0 ∂t ∂x ⎝ 2μ 3μ ∂x 3 ⎠

(A2)

We seek a solution of the form Δh = Dβ t βϕ(η)

(A3)

where η ≡ x/(Etα) and Dβ and E are constants that render ϕ and η dimensionless. Its validity in the region where the Marangoni stress is zero114−116 implies α = 1/4. At the boundary of the narrow region, the flux must be continuous, which implies that 3654

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h0 2τ h 3γ ∂ 3Δh ∼ 0 ⇒ t κ − α ∼ t β − 3α 3 2μ 3μ ∂x

STW, applied science division of NWO, and the Technology Program of the Ministry of Economic Affairs. The authors thank Jørgen van der Veen and Henny Manders for their help with constructing the experimental setup.

(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 t βϕ(η)] = Dβ βt β − 1ϕ + Dβ t β dη ∂t ∂t

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⎡ dϕ ⎤ = Dβ t β − 1⎢βϕ − αη ⎥ ⎣ dη ⎦ Dβ β − α dϕ dϕ ∂η ∂ [Dβ t βϕ(η)] = Dβ t β t = dη ∂x E dη ∂x 4 Dβ d4ϕ ⎛ ∂η ⎞ d 4ϕ ∂4 [Dβ t βϕ(η)] = Dβ t β 4 ⎜ ⎟ = 4 t β − 4α 4 4 dη ⎝ ∂x ⎠ E dη ∂x

At κ − 2α d ∂τ ∂ [At κδ(x)] = [δ(η)] = ∂x ∂x E 2 dη

where we have used δ(kx) = δ(x)/|k|. The ODE thus becomes βϕ − αη

2 3 dϕ 1 h0 γ d3ϕ ⎤⎥ d ⎡⎢ h0 A ( ) δ η + + =0 dη dη ⎢⎣ 2μDβ E2 E 4 3μ dη3 ⎥⎦

Setting E≡

4

h0 3γ 3μ

and Dβ ≡

h0 2A 2μE

2

=

A 3h0 4μγ

yields βϕ − αη

■

dϕ d3ϕ ⎤ d ⎡ + ⎥=0 ⎢δ(η) + dη dη ⎣ dη 3 ⎦

(A5)

ASSOCIATED CONTENT

S Supporting Information *

Supporting Information is available free of charge via the Internet at http://pubs.acs.org/. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00278. Video of a typical experiment. Data supporting the negligible difference between intermittent and continuous illumination. Further simulation results analogous to Figure 6 but using the surface-dominated reaction model. (PDF)

■

REFERENCES

AUTHOR INFORMATION

ORCID

A. A. Darhuber: 0000-0001-8846-5555 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge that this research is supported partially by the Dutch Technology Foundation 3655

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DOI: 10.1021/acs.langmuir.7b00278 Langmuir 2017, 33, 3647−3658