## Abstract

A model for the optical path difference introduced by a soap bubble in transmission is described. The model is then used with interferometric data to solve for the fringe order, and to define a procedure to extract the global film thickness in presence of turbulence flows occurring during the drainage process due to gravity. Experimental results on soap bubbles examined in single-pass phase-shift interferometry are presented.

©2013 Optical Society of America

## 1. Introduction

Soap films, bubbles and foams are of great relevance to life sciences and to industrial processes. Their basic structure consists of a liquid confined between two layers of surfactant material under the action of various forces such as surface tension and van der Waals attraction. Phospholipids in biological cells behave like surfactants in making up semi-permeable bilayers; plasma membranes and vesicles are elementary structures that can be modeled in a manner similar to soap films and bubbles. The nature and properties of such structures have been the subject of extensive studies [1–4], and continue to be attentively investigated, particularly in connection with the possibility of long-lasting permanence of humans and biological products in microgravity environments during space missions.

On Earth, a basic process observed in soap films and bubbles after formation is drainage of the inner liquid, and film thinning. The kinetics of the process generally involves hydrodynamic convection and results in a turbulent flow where thin regions go up and thick regions go down, often exhibiting plumes, trickles and streams. Eventually, if the film does not rupture, the structure ideally shrinks to a pair of facing layers of surfactant molecules (the so-called Newton black film, made of two monolayers, or the common black film, made of two multiple layers). A quantity of significance to study the physics mechanisms involved in the overall process is the actual film thickness, which is relevant to most parameters characterizing the interfacial rheological and structural properties of the system. In the case of vertical films, acting as thin planar plates, methods to measure the actual film thickness reported in the literature are x-ray reflectivity at grazing incidence [5–7] and visible light reflectivity at close normal incidence [8]. Differential interferometry methods have also been described in investigations of contact angles [9], bubble caps [10], and more [11–13]. A method based on phase shift interferometry was developed as well, and measurements on vertical films were demonstrated [14]; validation of the method was achieved by comparison with results obtained with polarization homodyne interferometry [15,16]. Other approaches were also reported [17–20], based on resonant differential interferometry [21] and on fringe patterns from a dual-wavelength reflection configuration [22].

As to soap bubbles, the knowledge of the film thickness plays a key role in the study of their dynamic interfacial properties, as it is the case for example in investigations by capillary pressure methods [23]. A measuring technique that has been recently proposed is by large lateral shearing displacement interferometry [24]. The thickness measurement, however, is particularly difficult owing to the curved shape of the film and to turbulent flows arising at drainage. While the latter are interesting in themselves (a recent study has taken soap bubbles as physical models accounting for thermal convection and emergence of isolated vortices in the atmosphere [25]), such flows constitute a perturbation of significance to interferometry. In order to obtain a global value of the bubble thickness free of turbulence perturbations, special techniques need to be devised and developed.. This in fact is the purpose of the present paper, where we present a measuring approach based on a model of the optical path difference (*OPD*) introduced by a soap bubble, and we describe a procedure of data processing that accordingly provides a value representing the film thickness of the bubble on the whole. The method we use also avoids a major inconvenience of standard interferometry approaches when dealing with interference patterns made of two unconnected regions, as it occurs with soap bubbles on a reference background. In fact in that case the integer number of wave lengths making up the gap between the bubble and the background is unknown, so that the customary determination of the *OPD* would be affected by intrinsic ambiguity.

A demonstration of the new measuring chain is given, and experimental results on soap bubbles examined with phase-shift interferometry are presented.

## 2. Geometrical *OPD* model of an ideal soap bubble

In this section we describe the geometry of a spherical bubble and derive a mathematical expression for the *OPD* map that is expected when such a bubble is passed by a light beam. Such an expression will be used in Section 4 to compute the bubble thickness from *OPD* data.

We refer to a soap (water) bubble in air, as it can be obtained from a vertical nozzle by transferring onto it a horizontal soap film with a frame and inflating from below. In the absence of perturbations, the bubble is modelled as a spherical film of uniform thickness; the region about the nozzle is disregarded, as well as the draining phenomena. In Fig. 1 we represent a sectional plane containing the bubble center, taken as the origin O of the coordinate system, and the z-axis singled out by the propagation direction of the light beam. We indicate with *R* the radius of curvature of the outer sphere of the bubble, *d* the film thickness and *n* the refractive index of water. Locally the film can be assimilated to a plane parallel plate made of a water layer. In normal section, the general case of a ray impinging on such plate at oblique incidence is illustrated in Fig. 1(a). The quantity of interest is the *OPD* defined as the difference between the optical path of a ray when the bubble is not there and when it is in place. Indicating with ${\vartheta}_{0}$and $\vartheta $ the angles of incidence and refraction, respectively, the *OPD* is expressed by

*d*is the film thickness.

The case of a ray intercepting the bubble at a particular position *(z,y)* is schematically shown in Fig. 1(b). The height *y* is normalized by the radius R yielding the normalized ray height $q=y/R$, with $0\le q\le 1$; due to axial symmetry, it is intended that the y-coordinate represents the radial distance of the incidence point from the z-axis in any sectional plane containing the z-axis. Naturally, *q* also represents the normalized radius of the *OPD* map. The incidence angle is ${\vartheta}_{0}=\mathrm{arc}\mathrm{tan}\left(\text{\hspace{0.17em}}q/\sqrt{1-{q}^{2}}\right)$, and the refraction angle is $\vartheta =\mathrm{arc}\mathrm{sin}\left[\left(\mathrm{sin}{\vartheta}_{0}\right)/n\right]$ . The single-pass *OPD* through the entrance and exit films of the bubble is therefore

*OPD*profile is a standard function $f\text{\hspace{0.17em}}\left(q,n\right)$ that scales times the geometrical thickness

*d*of the bubble film. Using for example the value

*d*= 100 nm,

*q*ranging from 0.0 to 0.8, and

*n*= 1.333, the resulting plot for

*OPD*(

*q*) is shown in Fig. 2. As a general notation, it is seen that the

*OPD*is always negative, meaning that the insertion of a bubble in the measuring arm of an interferometer produces an optical delay.

The mathematical representation of the geometrical *OPD* so far discussed does not take into consideration grazing incidence phenomena and guided light propagation occurring when *q* approaches unity; with soap bubbles in air, we conservatively limit our study to $q\le 0.8$ .

## 3. Pressure and temperature effects of the inner air on the *OPD* induced by a bubble

The geometrical OPD model described in Section 2 implicitly assumes that the air contained in the soap bubble is at the same pressure and temperature as the outer air. However, a departure from such conditions has to be expected, due to surface tension of the bubble film and to work performed to inflate the bubble. In this section the amount of such a departure is studied, to estimate whether it is relevant to the interferometric measurement of the bubble thickness.

For a soap bubble in air, after formation the air pressure and the temperature inside the bubble are higher than those outside. In principle, although such differences oppositely affect the air refractive index, the net contribution to the *OPD* is to be taken into account. Edlén’s formula [26] relating the pressure$P$ and the temperature $T$of air to the refractive index ${n}_{air}$ provides

The pressure difference $\Delta P$ is [27]

where $\kappa $ is the surface tension of the soap solution. Taking as typical values $\kappa $ = 3.0 10^{−2}Nm

^{−1}, $P$ = 1.0 10

^{5}Nm

^{−2}, ${n}_{air}-1$ = 2.8 10

^{−4}and $R$ = 1.5 10

^{−3}m, it is found that the contribution due to pressure increase is $\Delta {n}_{air,\Delta P}$ = 2.2 10

^{−7}.

As to the increase of temperature, it is assumed that blowing a bubble the air inside the bubble and the communicating reservoir at least initially undergoes a quasistatic adiabatic process, so that

with $V$the total volume (bubble plus reservoir), and $\gamma $ = 1.4 the ratio between the specific heat at constant pressure and that at constant volume of air; the subscript $A$ indicates the initial state when the bubble is still to be formed, and $B$ the final state with the bubble in place. The quantity${P}_{A}$ is known, then obtaining ${P}_{B}={P}_{A}+\Delta P$ by use of Eq. (5); ${V}_{A}$ is the volume of the reservoir only, which is known as well, so that Eq. (6) provides the final volume ${V}_{B}={V}_{A}{\left({P}_{A}/{P}_{B}\right)}^{1/\gamma}$. The work ${W}_{AB}$ done on the total volume of air to go from state$A$ to $B$ is^{−7}, and overall

*OPD*already computed with the geometric model of the bubble. The absolute value of the

*OPD*pertaining to a sphere of inner air has a maximum along a diameter; for the case in point we have $OP{D}_{\mathrm{max}}=2R\Delta {n}_{air}=-0.87\text{nm}$. Such a value is anyway smaller by an order of magnitude than the geometrical

*OPD*at

*q*= 0 that is measured in experiments with the thinnest soap bubbles, and can be safely neglected for bubbles not reaching the condition of black film. It is also considered that reaching the condition of black film generally requires several minutes. During this time, thermal exchanges through the soap film between the inner and outer air are occurring, tending to reduce and eventually annul the temperature difference. As a consequence, $\Delta {n}_{air,\Delta T}$ gradually goes to zero, so that only the small contribution due to $\Delta {n}_{air,\Delta P}$ is remaining.

## 4. Thickness computation

Real bubbles suffer from the effects of various perturbations, so that their shape somewhat departs from the ideal model of a spherical film with uniform thickness. As a magnitude of significance, however, one can refer to the *OPD* (*q* = 0), expressed by $-2\text{\hspace{0.17em}}(n-1)\text{\hspace{0.17em}}d$, thereafter computing the thickness *d* at the bubble centre. Expected values for *d* range from thousands or several hundreds of nm when the bubble is first formed, to a few tens of nm at the end of the thinning process [16]. Thick and slowly thinning bubbles are likely to be observed in microgravity environments, where draining might be considerably hampered. However, such a relatively large range poses intrinsic problems to the use of interferometry as a measuring method. In fact, processing the fringe pattern produced with transmission interferometry, two separately connected *OPD* regions are obtained, one of which corresponds to the bubble and the other to the surrounding area; the latter defines the reference optical path for the determination of the *OPD* (*q* = 0). The intrinsic inconvenience is that such two regions are not connected to each other, so that situations of ambiguity due to integer multiples of the wavelength λ cannot be excluded. For example, only considering a measured *OPD* of −66 nm at the center of the interference pattern as in Fig. 2, it is not possible to distinguish whether the true *OPD* is −66 nm, −66 nm ± λ, −66 nm ± 2λ or more, since all of them are within the expected *OPD* range. In practice, limiting the analysis to the bubble center with respect to the outside region, only the fractional part of the *OPD* (in λ units) can be reliably retrieved. In addition, only referring to the *OPD* at the bubble center, such a fractional part would also be severely affected by the disturbance due to turbulence.

To solve for the above problems, here we use the bubble model discussed in Section 2. As noted, the standard shape function *f* scales times the geometrical thickness *d* of the bubble film to yield the actual *OPD*. As a consequence, fitting the *OPD* map to the mathematical representation of *f* would directly produce a coefficient equal to *d*. However, as given in Eq. (3), such mathematical representation is somewhat involved for use with analysis programs. By numerical fitting with least squares techniques it is easily seen that a convenient polynomial expression for *f* (*q*, *n*) is

*n*= 1.333, limiting the range of

*q*to $0\le q\le 0.8$ and setting

*N*= 5 (so selecting even powers of the normalized radius

*q*up to

*q*

^{10}), best fit is obtained with the parameters listed in Table 1.

With the values in Table 1, the residuals of the fitting to the standard shape function given by Eq. (3) are limited to a maximum of the order of 1·10^{−5}. In case it is required, one might also express Eq. (3) in terms of radial Zernike polynomials, either repeating the fitting procedure or using appropriate conversion formulas [28].

In practice, to also single out the disturbance due to turbulence, we make a least-squares fitting to a set of azimuthal Zernike polynomials, as it is customary when analyzing interferograms in optical testing. We make use of polar coordinates with the polar axis along the x-axis and $\phi =\mathrm{arc}\mathrm{tan}\left(y/x\right)$ as the azimuthal angle. As to the radial polynomials, we only include the piston, plus a single term consisting of a truncated power series as given in Eq. (10), with coefficients *a*_{1}, … *a*_{5} accounting for the standard function as in Table 1. The set of fitting functions we use is then

*c*

_{i}, one for each fitting function of the set in Eq. (11); addressing in particular

*c*

_{4}, Eq. (2) directly providesThe pertaining uncertainty is given by the fourth diagonal element of the autocovariance matrix associated to the fitting. Such an uncertainty value shall then be combined with the uncertainty contributions from the data acquisition process (mainly, the uncertainty on the bubble diameter). As to the least squares fitting itself, we use standard routines available from general mathematics packages, just specifying the set of functions ${f}_{i}$ to be used as listed in Eq. (11).

Numerical simulations are consistent with the analysis given above. In particular, Eqs. (2) and (3) were used to generate synthetic data of soap bubbles of various thicknesses, embedded in a background of azimuthal Zernike terms and random (Gaussian) noise. Sampling was made on a Cartesian grid of points representing the pixels of a CCD camera. The data were then processed according to the procedure outlined above. Although the results are also influenced to some extent by the digitization depth of the simulated *OPD*, and by the maximum value of the normalized radius *q* that is considered, the compliance with the source thickness data is within the uncertainty given by the amount of Gaussian noise that was added. For example, simulating the *OPD* induced by a soap bubble with a film thickness of 100 nm and a Gaussian noise of 1%, the recovered thickness after processing is 100 ± 1 nm. The uncertainty in this case is very small because the data were generated by computer, and no uncertainty contribution from the bubble diameter was considered. Dealing with real data, where the diameter of the bubble is also to be determined, more contributions to the uncertainty balance as mentioned above shall be added.

## 5. Experimental results

The optical configuration we refer to is that of a Fizeau interferometer [29]. The main specifications of the instrument we use are listed in Table 2; the diameter of the output beam is reduced to 33 mm by means of an aperture converter.

The laboratory setup is schematically shown in Fig. 3. The transmission flat mounted on the mainframe of the interferometer is made of a slightly wedged plate with flat surfaces. One of the surfaces is anti-reflection coated, while the other is uncoated and provides a partial back reflection that serves as a reference to the probe beam exiting the interferometer. Such a probe beam passes through the soap bubble whose film thickness has to be determined, and is laterally displaced and back folded by means of a right angle prism. It then passes again through the transmission flat, and back-propagates into the interferometer superimposed with the reference beam above. The probe and the reference beams eventually make up the interference pattern at the detector array. Naturally, a similar path is also travelled by the part of the probe beam that first goes to the right angle prism, is laterally displaced and back folded, and then passes through the soap bubble, so that the interference pattern is made of a pair of equivalent images of the bubble, one on the side of the other. One of such images is arbitrarily selected, and the focus of the interferometer’s optics is adjusted for sharpness of the bubble edge. Interferograms are then acquired and processed.

The choice of the single pass configuration outlined above in place of a more sensitive double pass arrangement is imposed by the need of reducing the error on the determination of the bubble diameter; the latter intervenes for the normalization of the image coordinates during the stage of interferogram analysis and therefore affects the final result.

The interferogram of a typical soap bubble on a nozzle is shown in Fig. 4. To compute the bubble center and diameter, we refer to the no-data points produced by the interferometer about the edge of the bubble, where connection between the inner and outer regions is missing (Fig. 5); the bottom area where the bubble is attached to the nozzle is disregarded. Such no-points are used for a fitting to a circle, obtaining in particular the diameter of the bubble and the estimated uncertainty. In our case we find a diameter of 148 ± 1 pixel, corresponding to 5.17 ± 0.03 mm after calibration of the image space (34.9 μm/pixel). Conventional decoding with standard software provides the single-pass *OPD* map shown in Fig. 6. As anticipated, however, ambiguity effects due to the missing connection are showing up. In fact the bubble induces an optical delay, so that the central hemisphere should appear shifted downwards, below the level of the background. The interferometric data are therefore re-processed by singling out a central portion of normalized radius *q* = 0.8, and submitting it to the fitting with the set of functions given in Eq. (11). The resulting coefficient of the fourth function is then taken as the bubble thickness, according to Eq. (12). In our case such a thickness is 313 ± 14 nm; the stated uncertainty is estimated numerically, repeating the analysis with a bubble diameter at the extremes of its error range (147 and 149 pixels), and quadratically combining the half-difference of the resulting diameters with the uncertainty from the autocovariance matrix of the fitting mentioned above.

Although only within some approximation, the consistency of the measurement with the approach detailed in Section 4 is confirmed by a simple estimate of the bubble thickness from the *OPD* map given in Fig. 6. The *OPD* profile selected with a line through the central part of such a map is shown in Fig. 7. Using inspectors provided by the software of the interferometer, it is seen that the bubble occupies a lateral distance of 144 pixels. Taking such a distance as the bubble diameter, a reduction by a factor 0.8 leads to 115 pixels. Still working with inspectors, it is seen that the peak-to-valley of the central part of the profile for a width of 115 pixels is 88 nm. In Fig. 2 it is shown that the *OPD* difference between *q* = 0.0 and *q* = 0.8 for *d* = 100 nm is approximately 27 nm. The thickness estimate of our soap bubble is then (88/27)·100 nm = 326 nm. Although it is obtained using a single profile of the bubble instead of the entire bubble cap within *q* = 0.8, and also with some rough approximations, such a value is compatible with the result of 313 ± 14 nm computed with the general procedure described in Section 4.

## 6. Concluding remarks

The measurement of the film thickness of a soap bubble by phase-shift interferometry has been discussed, on the basis of a geometrical model of an ideal bubble. A processing method to overcome the ambiguity of integer multiples of λ in the reconstructed OPD map has been presented, also filtering out the major effects of turbulent flows within the soap film. The thickness that is derived is not the value at a specified location, but it is the result of a fitting operation over an entire area of the interferogram. As such, it is suited to represent the global value for the bubble thickness that was sought in the first place to characterize the bubble on the whole. The approach can be considered for experiments in microgravity conditions, where the setting and draining of the film that is observed on Earth might exhibit interesting new behaviours, of relevance both for physics and for life sciences.

## Acknowledgments

This work was performed under the auspices of the Italian Space Agency (ASI), within the project Liquid Film Tensiometer (LIFT). LIFT is an interferometry-based experiment designed to produce, measure and control very thin water bubbles in a fluid matrix in a microgravity environment. Its scope is to study the mechanisms to create and control emulsions and foams with a view to several application domains (chemistry, food, pharmacology, biomedics, and more). LIFT is expected to be flown on the International Space Station. The authors thank the Italian Space Agency for kind permission to publish this work. We are also grateful to C.M.C. Gambi for valuable comments and discussions on the subject of liquid films, and to S. Acciai and M. D’Uva for fabricating mechanical parts used in this study.

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