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In this paper we outline a new all-optical non-contact technique for measurement of the surface tension of a Newtonian fluid. It is based on the accurate measurement of the spacing of the supernumerary fringes produced by the diffraction pattern of a laser beam transmitted through or reflected by a thin vertically-draining film of the liquid. We discuss the basic theory and application of this technique, and several issues which must be addressed before it can be used commercially.
©2008 Optical Society of America
1. Introduction
The physics of vertically-draining liquid films was first investigated by Jeffreys in the 1930’s; he considered the shape of a Newtonian fluid attached to a vertical substrate as a function of time and two liquid parameters, density and viscosity. (This situation can be produced by dipping a glass slide into a beaker filled with the liquid and then lifting the slide vertically from the container relatively quickly). He was able to show that if one ignored surface tension effects, the liquid film would quickly assume a parabolic shape (the “Jefferys parabola”) which thinned over time.[1] However, the model which he developed ccould not be correct near the top of the fluid film, as it assumed that the slope of the fluid near the contact line would be horizontal - i.e., perpendicular to the substrate which the film was attached to. This is incorrect, as the fluid must approach the substrate at a “contact angle” determined by the relative surface free energies of the liquid and substrate.[2] In the early 1980’s, Tanner was able to improve on this model by including the effects of surface tension; in his model, surface tension effects dominate near the point of attachment of the fluid, while lower down the fluid shape approaches the Jefferys parabola.[3] He was able to show experimentally that this model was approximately correct for fluids which wet the surface of the substrate (i.e., had a contact angle of approximately 0°.) In this model, the shape of the draining fluid layer has a point of inflection due to the transition between the two regions. In this paper, we will show that the far-field light pattern scattered from the fluid layer near the point of inflection has a simple fold caustic (i.e., a “rainbow caustic”), and that the interference pattern produced by this caustic can be used to measure the surface tension of the fluid, potentially to high accuracy.
2. The tanner model
In this model, the shape of the draining fluid layer depends on three fluid parameters: the density, ρ, the surface tension, σ, and the viscosity, µ. The shape of the layer is given by the formula
Here, t is the time since the fluid layer was created, g is the acceleration of gravity, x is the horizontal thickness of the layer, and y is the vertical distance from the point of attachment. The constant k=0.455 is a dimensionless constant determined from numerical integration. [3] (We have written the two groupings of constants as A and B above for ease of theoretical analysis.) Figure 1 is a graph of the shape of the fluid layer for several different times; as one can see, the thickness of the film decreases over time, as expected.
Fig. 1. Profile of a draining fluid layer as a function of time. Fluid parameters: Surface tension, σ=0.02 N/m (20 dyne/cm); specific viscosity, µ/ρ=5×10-5 m2/s (50 cS); density, ρ=740 kg/m3 (0.74 g/cm3). A) t=0.001 s; B) t=0.01 s; C) t=0.1 s.
It can also be seen from Fig. 1 that the fluid layer has a point of inflection, where the curve goes from convex to concave. The coordinates of the point of inflection (x 0, y 0) can be determined by setting the second derivative of y with respect to x to zero; this yields
Near the point of inflection, the thickness of the fluid layer x(y) at a given point in time can be expanded to cubic order:
where and . Generally speaking, these coefficients β and δ can depend on time. We will now show by means of a scaling argument that δ is a constant, independent of time which depends only on σ and ρ. Before going on, however, we will stress that this is the central point of this paper, from which all else follows.
From Eq. (4), β must be dimensionless, while δ must have dimension L-2, where L indicates length. However, β and δ must be some combination of the coefficients A and B from Eq. (1). We will assume that the shape of the curve locally scales with time in the same manner that x0 and y0 do; that is,
Therefore, β~t -2/ 5 and δ~t 0. Because A depends explicitly on time, δ can only depend on B; because B has dimension L2/3, δ≺B -3≺ρg/σ.
One can also show this directly from calculation of the first and third derivatives; this can be done using two formulae from Abramowitz and Stegun [4]:
Using these, we can calculate β and δ:
Because g is a known quantity and ρ can be easily measured, measurement of δ can be used to measure the surface tension σ.
3. The rainbow caustic
If a plane wave moving in the x-direction passes through the liquid layer near the point of inflection, it will be phase-delayed; the relative phase will be a function of y:
Because of the point of inflection, the second-order term vanishes. The disappearance of the second-order term in the local expansion of the phase indicates the appearance of a fold, or rainbow, caustic in the far-zone diffraction pattern due to the thin film.[5,6,7] In terms of geometrical optics, the direction of the wave normals (i.e., “light rays”) goes through a local maximum, meaning that the rays seem to fold back on themselves (see Fig. 2).[6] (It is referred to as a rainbow caustic because the meteorological rainbow is due to a similar folding of light rays from the combined reflection and refraction of light by a water droplet.) Two light rays are scattered into a given angle below the maximum, while none are scattered at angles above the maximum (“rainbow”) angle; thus, one expects to see interference fringes due to the two-ray interference below the rainbow angle, and darkness above it. This is indeed seen in the meteorological rainbow, where for historical reasons the dark sky above the rainbow is referred to as “Alexander’s dark band”, and the interference fringes seen below the rainbow are referred to as “supernumerary rainbows.”[8,9]
Fig. 2. Refraction of a parallel bundle of light rays by the fluid layer near the contact line. Note the point of inflection leading to the maximum deviation of the light ray (indicated by a dashed line.) Units are normalized.
We will now give a semiclassical derivation of the interference fringes due to the thin film based on Fraunhoffer diffraction theory.[7,10] If we project the light diffracted from the liquid layer onto a screen whose vertical coordinate is Y, the electric field on the screen is proportional to the Fourier transform of the field just after propagating through the thin film:
in the paraxial limit. (We ignore reflection and transmission coefficients due to the thin film.) It is straightforward to show that in this approximation the far-zone electric field is proportional to an Airy function, and thus the intensity to the square of the Airy function:
Here, the angle θ=Y/L, and θ0 is the rainbow angle (=(n-1)β in the paraxial limit); the constant K includes all of the physical constants neccessary to make the units come out right. Figure 3 shows a graph of the Airy function squared; a number of reference books and websites carry routines for calculating the Airy function, its zeros and extrema, or tables of the same.[11,12] The first column of table 1 shows the first five maxima of the Airy function squared, taken from table 10.13 of Abromowitz and Stegun.[13]
Fig. 3. Graph of the function v=(Ai(z))2. Note that the function decays without oscillation for z>0, while it oscillates (i.e., displays interference fringes) for z<0.
Table 1. Positions of Airy Function Maxima and Fringe Maxima
The spacing of the maxima of the interference fringes is determined entirely by the cubic term, δ, of the phase; however, δ is time-independent and determined by the surface tension, σ, of the liquid. Hence, the accurate determination of the spacing of the interference fringes can be used to determine the surface tension of the liquid. If one measures the positions of the maxima of the interference fringes, one can determine the surface tension from
Here, θi is the angular position of the ith supernumerary peak (treating the main rainbow fringe as the 0th peak) and zi is the ith maxima of Ai(z)2 (labelling the first maximum as z0).
4. Rainbow surface tension analysis
We have made a “proof-of-principle” measurement of the surface tension of a fluid from the rainbow caustic created by shining a laser beam through it. The experimental setup is shown in Fig. 4; we shine the HeNe laser (λ=632.8 nm) through the glass slide, and raise the beaker filled with the liquid so that the contact line of the liquid is at the point where the laser beam intersects the slide. We then remove the beaker quickly; the diffraction pattern due to the thin liquid film is projected onto a ground glass screen, and is recorded by a video camera. The setup is very similar to one used in a previous paper to record the twin-rainbow pattern created by a thin liquid film coating a glass rod. [14,15]
Fig. 4. Experimental setup to measure interference fringes from the draining liquid layer. A) Experimental layout as seen from above. B) Profile of slide and draining layer.
The substrate which the liquid film drained on was a pre-cleaned microscope slide. No attempt was made to clean the slides further than what had already been done, and we discarded slides after one use. We did not attempt to account for the refraction due to the slide when calculating the surface tension; for reasons discussed below, corrections due to the slide refractive index are second-order effects. The liquid used is a 50 cS viscosity silcone oil with a nominal surface tension of 20 dynes/cm (0.02 N/m). It is the same liquid as used in the paper referred to above. Because the rainbow angle decreases over time, the camera sees an upwardly-moving interference fringe which sweeps by the camera on a timescale dependent on the fluid viscosity. For the fluid being used here, the fringe pattern takes of order 2-3 s to sweep past the camera. One technical point: because the substrate has a liquid film on both sides, the phase change due to the film is double that from Equation 6; this is accounted for in the formulae which follow.
Figure 5 shows an image of the interference fringes of the rainbow caustic; it was taken 2.3 s after removing the substrate from the liquid. Figure 6 shows the intensity of light near the rainbow angle digitized from Fig. 5; the positions of the first five peaks of the interference fringe (0 through 4) are labelled on the figure. The position of these peaks was determined by eye.
Fig. 5. Image of rainbow fringes due to scattering from the fluid layer. The different supernumerary peaks are labelled on the figure.
Fig. 6. Digitized intensity of the interference fringes of fig. 5. Pixel number increases toward the bottom of the image (see Fig. 5). Supernumerary peaks are indicated on the figure.
Using Eq. (13), we can calculate the surface tension of the liquid using the position of the first several supernumerary rainbows. The third and fourth columns of Table 1 shows the angular separation of the “zeroth” peak to the ith peak, and the surface tension calculated using these data. From this, we find an average value of σ=22.75±3 dyne/cm; the nominal value of σ for this fluid is 20 dyne/cm, indicating that the technique works.
It should be mentioned parenthetically that Tanner measured the rainbow caustic due to a second point of inflection at a lower position (the “head” region) on a draining fluid layer in a 1978 paper. He was able to analyze the effect of surface tension on the shape of this section of the droplet using the rainbow caustic; however, in that case the cubic term (δ) was not constant for reasons given in the paper.[16]
5. Accuracy and limitations of RASTA
The experimental data taken above indicate about a ±15% spread in the value of the surface tension as measured using RASTA. This spread is most likely due to the limitations of the camera which we used for the “proof-of-principle” measurements. The camera which we used was a standard CCD video camera with a vertical resolution of 480 pixels. As can be seen from Fig. 6, the spacing between adjacent peaks was of order 30-80 pixels; taking 50 pixels as an “average” separation, and assuming an uncertainty of ±1 pixel in locating the peak of the interference fringe, we have a relative uncertainty in peak separation of ±2 pixels/50 pixels ~ 4%. Because the RASTA value of the surface tension is proportional to the cube of the peak separation, the uncertainty in the surface tension due to camera resolution is of order 3 times the uncertainty in peak separation, or of order ±12%, which is about the spread in measurements of the surface tension. We are in the process of replacing the camera with a high-resolution (2048 pixel) linear array camera, which should in principal allow us to reduce the measurement uncertainty by a factor of 4 or better; in addition to this, we are also developing computer routines to allow us to determine the peaks to sub-pixel accuracy, which should improve the measurement uncertainty by another factor of 2. This will allow us to measure the surface tension to an accuracy of roughly ±1.5% or better using the new camera system.
Fundamental issues probably limit the accuracy of these measurements to about the 1% level as well. As seen in Fig. 6, peaks 3 and 4 are split, due to a weak interference fringe on top of the Airy fringe; this was probably because of the weak internal reflection between the liquid and the glass substrate. These were not counted as two separate Airy fringes because of our knowledge of the spacing of the maxima of the Airy function; however, extraneous interference fringes such as these are probably unavoidable unless the substrate is index-matched to the liquid, and cause uncertainty in measurement of the peak of the fringe. Another issue is the accuracy of the Tanner model (Eq. (1)) in describing the shape of the fluid at short times, and the insufficiency of the Airy approximation to describe light scattered from the liquid layer at long times. The latter reason is why we used only 5 maxima of the fringe pattern to determine the surface tension, as we do not expect the high-order fringes to follow the Airy approximation closely; this is discussed in detail in section 6.
The Tanner model for the shape of the draining fluid is an approximation which assumes a 0° contact angle between the fluid and the substrate. This is a good approximation for the silicone oils which we used in this proof-of-principle, where the contact angle is about 1–2°; this does imply that this technique (at least, as currently implemented) can only be used for surfaces where the contact angle between the fluid and substrate is near 0°. It may be possible to use a roughened platinum surface for the substrate, and view the diffraction pattern in reflection, to get a 0°-contact angle for most liquids; we are currently investigating this possibility. In addition, the model is based on an approximation which is most valid for times long (compared to τ) after the fluid layer is established, and does not include any inertial effects due to the removal of the substrate from the fluid, which means that the fluid layer is probably not well described by the Tanner model at very short times. However, the Airy approximation used to derive Eq. (13) becomes less good for long times (compared to τ (a natural timescale defined in Eq. (14))) for reasons discussed in the next section. These considerations imply that there is a window of time (between about 1 τ and perhaps 100 τ) in which the surface tension measurements must be made, or at least that if measurements are made at longer or shorter times, more sophisticated modeling must be done to interpret the data. We are beginning to model the intensity of the light scattered by the fluid layer at long times (when the Airy approximation becomes invalid); this is the subject of the next section of the paper. One unexpected benefit stemming from this analysis is that we can use the failure of the Airy approximation to determine the viscosity of the liquid.
6. Computer modeling of light scattering from the liquid film
As shown above, the coefficient of the cubic term of the phase in light scattered near the point of inflection is independent of time. In the semiclassical theory this implies that the intensity of light transmitted through the thin film should be proportional to the square of an Airy function. [11] However, this is only true under two conditions: first, because the Airy integral is defined from +∞ to -∞, one needs to be able to ignore boundary effects; and two, one must be able to ignore the higher-order terms in the expansion of the phase.[7] For our fluid surface, both conditions are approximately true for short times after the draining layer was created but not for long times. One way to see that this must be the case is that at long periods of time, the angular position of the primary rainbow peak decreases over time (toward the projection of the line of attachment in the far-field); if the angular spacing of the rainbow didn’t decrease, the diffraction pattern would extend beyond the line of attachment at long times.
We have simulated the scattering of light by a thin liquid layer draining over time; this work is still in its initial stages, as our model simply simulates light scattering in the Fraunhofer regime by Fourier transforming the phase of a plane wave propagating through the fluid layer.[10] However, the preliminary results from this simulation are shown in Fig. 7, Fig. 8, and Fig. 9.
Figure 7 shows the intensity of light scattered by the layer at time t=0.07 s after the fluid layer was created. The parameters used for this simulation are σ=20 dyne/cm, µ/ρ=10 cm2/s, ρ=0.974 g/cm3 and n=1.4. (It is interesting to compare Fig. 7 to Fig. 6 and Fig. 3.) At this time, the spacing of the maxima is well approximated by the maxima of the Airy function, as will be shown below.
Figure 8 shows the position of the primary rainbow fringe as found from these simulations as a function of time. Time on the graph is normalized in terms of a natural timescale
Using the parameters given above, τ=0.07 s. As is seen by the curve fit, the position of the primary rainbow scales as t -2/5, as derived above analytically. Note that the position of the primary rainbow changes by over an order of magnitude over the time considered.
Fig. 9. Angular separation of interference fringe maxima (θ0i)Dots: Data from computer simulation Solid lines: Separation determined using the analytic approximation (Eq. (12)).
Finally, Fig. 9 shows the angular separation of successive maxima from the zeroth maxima (as defined above) in these simulations as a function of time. Note that although the position of the primary rainbow changes by an order of magnitude, the angular separation of the diffraction pattern maxima changes by very little, and the amount of change depends on the diffraction order, with the separation of higher-order maxima decreasing more than lower-order ones.
7. Future work
While preliminary work shows that this technique is probably useful for accurate determination of the surface tension of a transparent fluid, a good deal of work (both theoretical and experimental) has yet to be done. This includes:
Comparison of experimental data to simulation: The preliminary data shown above indicate that this technique will work, and that the data taken match the theory reasonably well. However, we have not yet made a detailed comparison of data taken over a long time period to the computer simulations of the light scattering shown above.
Using RASTA for Viscosity Determination: There are two potential methods for doing this. The first relies on the fact that the geometrical rainbow angle, θ0 changes over time as:
The position of the maximum of the primary rainbow peak is shifted slightly from the geometrical rainbow, but it should still scale as t -2/5 power. From this, we can use the position of the rainbow peak as a function of time to determine the viscosity of the liquid. Doing this accurately relies on changes to our experimental apparatus described below.
The second method is a novel approach to viscosity determination which relies on the results of the computer simulations of light scattering discussed above. As shown in Fig. 9, the spacing of the supernumerary rainbow peaks depends on time, although the rate at which the peak separation decreases is not uniform: the angular separation of higher-order peaks decreases before that of lower-order peaks does. The timescale over which this occurs can be seen from Fig. 9: at t=100 τ, θ05 has decreased from 0.035 rad to 0.028 rad, while θ01 has stayed essentially constant. One can therefore define a characteristic timescale over which the spacing of one of the interference maxima will decrease by a set amount (say, 20%) from its short-time value, and use this measurement, or a set of measurements on different fringes, to determine τ. (The characteristic time also depends on the index of refraction of the liquid, although because the index will vary from 1.2 to 1.5 for most liquids, the index dependence is minor and can be taken into account easily in theory.) Therefore, the short-t value of the fringe spacing can be used to determine σ, and the long-term variation in the fringe spacing can be used to determine τ, and hence µ.
Comparison of measurements made via RASTA to other techniques: One of the biggest weaknesses of this paper is that we have not made any direct comparisons of measuremens of the surface tension made by RASTA with other,more conventional, approaches. We are in the process of acquiring a tensiometer to make these comparisons.
Analysis of substrate surface quality and cleanliness: To date the substrates which we have used are pre-cleaned biology cover slides. We have not worried about surface cleanliness or the optical quality of the substrate. To exaine this, we have purchased several high-quality optical flats which we are experimenting with, to use in our RASTA experiments. We are also using them to determine the effects of repeated cleaning (using sodium hydroxide) on the optical quality of the surface.
Automation of data taking and analysis: We are currently designing a computer-controlled stage to automatically lift the substrate out of the fluid and to trigger the camera. We are also in the process of replacing our current video camera with a high-resolution, high-speed line-scan camera to collect data, and are writing a program to automatically fit our data to an Airy function to calculate surface tensions. One issue with this is that there is significant dispersion in the value of σ as calculated in Table 1; we believe that by using a computer to fit the data, rather than determining the peaks “by eye”, and by using a high-resolution camera, we will reduce this dispersion significantly. Automating the timing of the experiment will also allow determination of the viscosity of the liquid by the two methods discussed above.
Contact angle measurement: Finally, in principle this technique should allow determination of the contact angle of the fluid attached to the substrate. Because in reality the fluid surface does not join the substrate at zero angle but at a finite contact angle, the refraction of light at very long times by the fluid layer should allow measurement of small contact angles.
8. Conclusions
In this paper we have demonstrated a “proof-of-principle” of a new optical technique to determine the surface tension of a Newtonian liquid. The technique is experimentally straightforward, although the analysis of the data relies on a detailed knowledge of the shape of the fluid layer and the theory of Fraunhoffer diffraction. The principal author of the paper (Dr. Adler) is in the process of applying for a patent for this technique (with the help of St. Mary’s College), which we feel is straightforward enough to have industrial applications.
Acknowledgments
This work was supported by Research Corporation grant # CC6308. We would like to thank Jim Lock at Cleveland State University for checking over our calcuations, and Simon Read of St. Mary’s College for several discussions about automated data analysis.
References and Links
1. H. Jeffreys, “The drainage of a vertical plate,” Proc. Cambridge Philos. Soc. 26, 204–205 (1930). [CrossRef]
2. P. C. Hiemenz, Principles of Colloid and Surface Chemistry, 2nd. ed. (Marcel Dekker, New York, 1986), pp. 287–299.
3. L. H. Tanner, “The surface tension effect on the flow of liquid down vertical or inclined surfaces,” J. Phys. D 13, 1633–1641 (1980). [CrossRef]
4. M. Abromowitz and I. A. Stegun, “Handbook of Mathematical Functions,” (Dover Publications, New York, 1970), pp. 16 (formula 3.6.25).
5. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc.6, 397–403 (1836), reprinted in P. L. Marston (ed.), Geometrical Aspects of Scattering, SPIE Milestone Series, vol. MS87 (SPIE, Bellingham, WA, 1994) pp. 298–309.
6. T. Poston and I. Stewart, “Catastrophe theory and its applications” (Dover Publications, New York, 1978) pp. 252–255.
7. J. F. Nye, “Natural Focusing and Fine Structure of Light” (Institute of Physics Publishing, Bristol, 1999) pp. 123–136.
8. R. Greenler, Rainbows, Halos and Glories (Cambridge University Press, Cambridge,1980), pp. 8–10.
9. M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 195–197.
10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57–74.
11. M. Abromowitz and I. A. Stegun, “Handbook of Mathematical Functions,” (Dover Publications, New York, 1970), pp. 470–478 (formula 3.6.25).
12. A good online resource for Airy functions is: http://mathworld.wolfram.com/AiryFunctions.html.
13. M. Abromowitz and I. A. Stegun, “Handbook of Mathematical Functions,” (Dover Publications, New York, 1970), pp. 478 (Table 10.13).
14. C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001). [CrossRef]
15. C. L. Adler, J. A. Lock, I. P. Rafferty, W. A. Hickok, and B. R. Keating, “Twin-Rainbow Metrology I: Measurement of the Thickness of a Thin Liquid Film Draining Under Gravity,” App. Opt. 42 (2003) 6584–6594. [CrossRef]
16. L. H. Tanner, “A study of the optics and motion of draining oil drops,” Opt.Technol. (1978) 125–129. [CrossRef]
