Implementation of an Axisymmetric Drop Shape Apparatus using a Raspberry-Pisingle-board computer and a web camera

Marcello Carla∗ and Antonio OrlandoDepartment of Physics and Astronomy, University of Florence,

Via G. Sansone 1 50019 Sesto Fiorentino - (FI) - Italy(Dated: January 26, 2018)

This paper describes the implementatin of an Axisymmetric Drop Shape Apparatus for the mea-sure of surface or interfacial tension of a hanging liquid drop, using only cheap resources like acommon web camera and a single board microcomputer. The mechanics of the apparatus is com-posed of stubs of commonly available aluminium bar, with all other mechanical parts manufacturedwith an amateur 3D printer. All the required software, either for handling the camera and takingthe images, or for processing the drop images to get the drop profile and fit it with the Bashforthand Adams equation, is freely available under an open source license. Despite the very limited costof the whole setup, an extensive test has demonstrated an overall accuracy of ±0.2% or better.

I. INTRODUCTION

Often in the study of fluids, in elementary physics, itis common to start with the statement “... neglectingsurface effects ...”. Probably, this is a necessary pledgeat the introductory level, but if this assumption is notreleased at a second stage, a physicist may miss in the realworld the physics of drops and bubbles, foam, cream, theformation of clouds in a storm, and many other things.

In the history of science, surface effects (also calledcapillary effects from the first observations of the phe-nomenon) have been a very important argument, becauseit was their presence to give, in the XVII and XVIII cen-tury, a first hint to the existence of forces other thangravity governing the state of condensed matter. Thesubject, already mentioned by Newton,1 was studied byYoung2 and Laplace.3 An exhaustive historical review ofthe foundations of capillarity, as well as a still valid in-troduction to the argument, is in Maxwell’s article inthe year 1876 edition of Encyclopaedia Britannica4. Inthe words of Maxwell, quoting Poggendorf, Leonardo daVinci was to be credited as the first discoverer of cap-illary phenomena. A more recent historical review canbe found in Ref. 5, as well as in the introduction to Is-raelachvili’s or Rowlinson’s books.6,7 Interfacial tensionwas also the argument of Einstein’s first published work.8

When the starting assumption is not made and thepresence of a surface is explicitly considered, in the workof a thermodynamical system a term dw = γdA has to beadded. The constant γ, with dimension energy/surfaceor, equivalently, force/length, is the interfacial (or sur-face) tension9 and accounts for the free energy changein the system when a new surface dA is created mov-ing molecules form the uniform bulk phase to the newenvironment that is the frontier marking the separationbetween the phases, the interface.

In the study of the properties of liquid surfaces orinterfaces, γ has been for a long time the most signifi-cant experimental quantity and most of the experimen-tal work aimed at measuring it as a function of temper-ature T , pressure, chemical potential µi of the species

composing the liquid phase(s) and, in the case of anelectrically charged interface, the polarizing potential E.Actually, such a knowledge allows through the capil-lary adsorption equation10,11 to obtain the surface ex-cesses Γi = −dγ/dµi, ΓS = −dγ/dT and σ = −dγ/dE.These quantities express the difference among an idealsystem, in which the two contacting phases extend ho-mogeneously up to a geometrical dividing surface, andthe corresponding real system, where in a thin but finiteregion there is a continuous transition from the proper-ties of one bulk phase to the other.

Hence, surface excesses Γi are a measure of the ten-dency for a substance to accumulate in the surface orinterfacial region, namely its tendency to act as a surfac-tant. The entropy surface excess ΓS is a measure of thelevel of greater or lesser molecular structuration carriedinto the system by the presence of the interface. Finally,charge surface excess σ is the surface charge density inthe electrical double layer formed by the two opposingphases, a quantity of paramount importance in electro-chemistry and in the study of the stability of colloids.

Surface tension and surface excesses are purely ther-modynamic quantities. Today, new techniques can givemore direct informations about binding energies, or thecomplete force law of the molecular interactions, or thestructural arrangement of molecules. Among these, wequote small angle x-ray, neutron or ion beam scattering,scanning and atomic force microscopies, etc. Any of theseterms, dropped into a web search machine, will give anidea of the wide variety of available resources.

Yet, according to Israelachvili about surface tensionand contact angle measurements, ”... these relativelysimple experiments can provide surprisingly deep insightinto the state of surfaces and adsorbed films”.6

Along the time, many techniques were devised to mea-sure interfacial tension. The most widely used were thedetachment methods (DM),12,13 the capillary rise (CR),4

the drop time (DT),14,15 and the maximum bubble pres-sure (MBP).16 All of these suffered either from being notsuitable for absolute measurements (MBP, DT)17, requir-ing calibration data to be obtained with some other ab-solute technique, or being dependent upon the presence

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of a third phase, extraneous to the studied interface, thatmight affect the results: the glass of the capillary for CRor the metal for DM.

It was known from Young and Laplace works2,3 thatthe shape of a resting liquid drop is modeled by surfaceor interfacial tension γ, by gravity acceleration g and bythe difference D1 − D2 beween the density of the dropliquid and its surrounding fluid.

On this basis in 1883 Bashforth and Adams publishedtheir book An attempt to test the theories of CapillaryAction,18 applying Young-Laplace theory to a liquid dropendowed with axial symmetry. The resulting differen-tial equation that describes the profile of the drop (theBashforth-Adams equation) has no known analytical so-lution, but can be integrated numerically. The book con-tains many tables of profile data obtained this way andhas been for about one century the fundamental referencefor the argument.

The inverse operation, namely obtaining from pointstaken on a drop profile the underlying differential equa-tion parameters, hence interfacial tension γ, was consid-ered a very interesting measure technique, because it isnon invasive, can yield absolute data, there is no need totouch the drop, but only to see it (hence it is possibleto perform also measurements under pressure or at hightemperatures). For all these reasons the drop shape mea-surements have always been considered to be the mostreliable and dependable ones for interfacial tension. Butthe procedures were not that simple.

Basing on Bashforth and Adams tables many methodswere devised. But they did find a very limited appli-cation for a long time because of the cumbersome workrequired to get and process experimental data. The dropprofile had to be taken manually, with a cathetometer ora travelling microscope, and computations to obtain fromexperimental data the value of γ were prohibitive in thelack of any automated computational resource. Due tothese difficulties, these methods were based on selectedfeatures of the profile, e.g., the height and diameter ofthe drop equator, hence yielded a limited accuracy. Areview of these methods can be found in Ref. 19, togetherwith a new graphical procedure for the interpolation ofBashforth and Adams tables and computation of interfa-cial tension fitting all points over the entire drop contourinstead of selected points only. This was a great step to-wards a greater accuracy, because the whole informationcontained in the drop profile was used; but this procedurealso was heavy.

As a fact, almost one century after Bashforth andAdams work, there was in the literature only one paperwith a series of measures of interfacial tension as a func-tion of polarization (an electrocapillary curve), contain-ing 27 values of γ at different potentials.20 This measure,made on the most studied model of polarized interface,a drop of mercury in a KCl electrolytic solution, had thepurpose mainly to yield calibration data to be used withother non absolute techniques.

The drop shape technique was very far from being suit-

able for routine work, but things were going to change,rapidly and deeply in a few years. First, the availabil-ity of computing power at an affordable cost made itpossible to fit the profile obtained from Bashforth andAdams equation over a full experimental profile usingan automated iterative numerical procedure.21 Second,it became possible to automatically acquire a full dropprofile in a single shot using a computer interfaced videocamera. Initially video camera tubes were used,22,23 butas soon as it was possible tubes were replaced by solid-state devices.24–29 Traditional videocamera tubes had ahigh level of geometrical distortion and, yet worse, in-stability in the magnifying factor, that made them un-suitable for accurate measurements. Solid state devicegeometry error, instead, is by far a negligible contribu-tion to the overall measure error.

This evolution transformed the drop shape techniquefrom a niche calibration procedure into a daily measure-ment routine, that was known as Axisymmetric DropShape Analysis (ADSA).

The single limit was the cost of the new technique.Soon commercial apparatus became available, but any-way the budget was of the order of several ten-thousanddollars.

Nowadays prices both for computing power and a highresolution solid-state imaging device have dropped bythree to four orders of magnitude and it is rather com-mon to see a web camera used in the implementation ofsimple scientific instruments (see, e.g., Ref. 30 and 31).

But this alone may be not enough because an ADSAapparatus requires also a suitable software to take andprocess the drop image and a mechanical and opti-cal setup that usually can be provided only in a wellequipped laboratory. The first point is not a large hin-drance, thanks to the widespread availabilty of opensource code either for commercial camera handling andimage processing,32 and for the technical algorithms fordrop profile detection and mathematical fit with Bash-forth and Adams equation.21,33,34

So we explored the possibility to overcome the last dif-ficulty making use of a new resource now available at abudget price for every experimentalist, namely the pos-sibility to assemble relatively accurate mechanical partsusing the 3D printing technique. We decided to imple-ment an ADSA apparatus of good enough accuracy toreplace the traditional detachment method most oftenused in classroom experiments,35 using only the resourcesavailable in most non-specialized laboratory at a budgetprice.

The result is the apparatus we are going to describe,assembled with a 19$ web camera and a 35$ Raspberry-Pi single board computer, with mechanical parts printedwith a 150$ 3D printer.

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II. THE DROP PROFILE

Currently, ADSA is a well established technique andwe refer to the previously quoted literature for any detail.In this paragraph, for the reader facility, we report inshort the main equations along the guidelines of Ref. 33.

The Young-Laplace equation

γ

(1

R1+

1

R2

)= ∆P (1)

states the link among interfacial tension γ, surface cur-vature given by R1 and R2 and pressure difference ∆Pbetween the two sides of the interface at any point Q ofthe surface.

Radii R1 and R2 are any two curvature radii obtainedcutting the surface at point Q with two normal sec-tions perpendicular to one another (it can be shown that1/R1 + 1/R2 is invariant on the choice of the sectionpair).36

In the case of an axially symmetrical drop, in thedrop apex at Z = 0, the curvature radii have identi-cal value (R1 = R2 = R0), and the pressure difference is∆P0 = 2γ/R0. In this conditions, from eq. 1 Bashforthand Adams equation ensues:

γ

(1

R+

sinϕ

X

)=

2γ

R0− gZ(D1 −D2) (2)

where ϕ is the angle between the Z axis and the externalnormal n to the profile at point Q of coordinates {X,Z},R is the radius of curvature at Q in the meridian planeand D1 and D2 are the densities of the drop liquid andof the surrounding fluid.

FIG. 1. Shape of a hanging drop. The dotted line is anarc of circumference, of radius r, tangent to the drop profile(solid line) at point Q. Vectors ds and n are respectivelya profile element and the normal to the profile at Q. In thetop-right inset, their decomposition into dx and dz cartesiancomponents, as given by eq. 5 and eq. 6.

Switching to reduced coordinates, x = X/R0, z =Z/R0 and r = R/R0 (fig. 1). Introducing the form factorβ = g(D1 −D2)R2

0/γ, eq. 2 becomes

1

r+

sinϕ

x= 2− βz (3)

whose solution is governed solely by the shape factor β.Eq. 3 can be solved by splitting it into the differential

system

dϕ

ds= 2− βz − sinϕ

x(4)

dx

ds= cosϕ (5)

dz

ds= sinϕ (6)

and numerically integrating, e.g. through a Runge-Kuttaprocedure37 (ds is the reduced profile integration step,see the top right inset in Fig. 1).

Obtaining from a drop image the form factor β andthe apex curvature radius R0, interfacial tension can becalculated inverting the expression for β given above:

γ = g(D1 −D2)R2

0

β(7)

III. EXPERIMENTAL

The mechanical arrangement of the ADSA apparatusis shown in the schematic plot in Fig. 2.

The solid state video camera, on the left, sees the pro-file of a resting liquid drop immersed either in a gaseousatmosphere or in another (transparent) liquid, and illu-minated from behind by a uniform light source for bestimage contrast.

It is well known that in ADSA technique best resultsare obtained with parallel light rays, that form a silhoutteimage of the drop. Usually this is obtained through a col-limating lens and a pinhole (see Ref. 38 for a detailed dis-cussion). We have found by trial that sufficiently goodresults can be obtained more simply using a uniformlyilluminated background made with a polycarbonate opa-line window at a suitable distance between the cell andthe light source (on the right in the plot).

There is no problem if the drop liquid is transpar-ent: the profile edge will appear black anyway and thickenough, because of refraction by the surface curvature.Light passes through only in the central part of the drop.This is shown in the inset, that reports the image of awater drop in air.

A. Optics

There are two strict requirements for the camera.First, it must be possible to read the image frame in rawformat. Every lossy compression algorithm introduces

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FIG. 2. Mechanical arrangement of the ADSA apparatus. Blocks marked with S are sleeves manufactured as short squaretubes, with dimensions carefully adjusted to fit onto the aluminum beams and slide softly back and forth. This allows an easydistance regulation among camera, lens, cell, opaline window and light source. Each sleeve is endowed with threaded bores onthe lateral and bottom faces, to secure with plastic screws the position after adjustment. The optical paths from camera tolens and from lens to cell can be protected from stray light with a dark plastic tube (not shown in the figure).

artifacts near a sharp contrasted bord, exactly where weneed the best of image accuracy and cleanliness. Second,it must be possible to firmly secure and align the cam-era on the apparatus with a mechanical stability at thelevel of one or at most few tenths of millimiter. These re-quirements rule out many commercial cheap cameras foramatorial photography, many webcameras with a USBinterface, and other resources like, e.g., a smartphonecamera.

We have used a web camera, purchased fromWaveshare,39 equipped with an OV5647 sensor with 2592× 1944 pixels and with adjustable focus. It was con-nected to a Raspberry Pi (B) single board computer40

with a debian distribution Linux operating system,41

used either standalone, with a screen, keyboard andmouse, or remotely from a PC.

At first, the camera was used with its original opticsand an high level of geometrical distortion was present.Because of the short original focal length, it was neces-sary to focus the small drop at a very short distance.A thin plumb line, used to align the bench, appearedstraight in the image when positioned in the meridianplane of the drop at middle of the frame. When movedtowards the sides of the frame, instead, it deviated fromstraight as much as 15 pixels. Despite this, operatingwith proper care and with some effort, valid results couldbe obtained, as explained in Section III-E.

Subsequently, to simplify the apparatus operation andimprove reliability and accuracy, the original optics wasreplaced with a lens with focal length f ' 8 cm and di-ameter 23 mm, with an 8-10 mm aperture. No deviationfrom the straight line could be detected any more in thewire image at any position in the frame. An achromatic

doublet lens was used, because it was at hand, but asimple lens should work as well.

The camera was used both at its highest resolu-tion (2592x1944) or in binning/subsampling mode, with1296x972 resolution, with no appreciable difference inmeasurement accuracy.

B. Mechanical assembly

As shown in Fig. 2, all parts of the apparatus are sup-ported by two beams of aluminium square tube with a 30mm × 30 mm cross section and 2.5 mm thickness, easilyavailable at many hardware stores.

The two beams, of lenght 100 cm and 65 cm, are jointedby double sliding sleeves (short square tubes, marked Sin the figure), manufactured to fit and softly slide backand forth on the beams and endowed with screws to blockmovement after adjustment.

The light source (a led work lamp mod. Jansio fromIkea) and the opaline window are directly mounted onthe longer beam with two simple sliding sleeves. Othercomponents, i.e. the measuring cell, the lens (when used)and the camera are mounted on the shorter beam. Allthese parts also are supported by sliding sleeves for po-sition adjustment, with screws for securing.

The lamp holder is a simple vertical bar, with a holewhere the lamp is secured by three screws oriented 120◦

each other. The camera holder is made of a pair oftwin sleeves and bars. The bars are provided with fourthreaded columns where the camera is screwed (only theside facing the cell being actually used).

The measuring cell is mounted on the open side of the

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FIG. 3. From left to right, details of the web camera assembly(without the optics), the measuring cell and the light source.

upper beam, to allow an easy removal for both mainte-nance and alignment with a plumb line to be placed inthe plane of the drop meridian. It is composed of a mainbox, a cover that fits tightly on the top and two framesscrewed on front and rear. The cover hosts a circularhole where the dropping tip (e.g., a syringe) can be fit-ted. The front and rear frames secure two microscopeslides used as clear windows along the optical path, tokeep the cell tightly closed and reduce evaporation. Abeaker in the lower part of the cell collects the dropsthat detach from the tip. Fig. 3 shows photo images ofthe main components.

As usual, the apparatus is supported by three levelingscrews, a single one at the left extremity of the mainbeam, the other two at the extremities of a third smaller30 cm long beam, mounted transversal to the main one atthe right extremity. The third beam also is an aluminimusquare tube, with a 15 mm × 15 mm cross section and 1mm thickness. The leveling screws end in a cap nut andare held in place by three small plastic cups fixed to thetable top with biadhesive tape.

All the mechanical parts, namely the measuring cell,the holders for lamp, lens, opaline window and cameraas well as all sliding sleeves have been manufactured us-ing a Prusa i3 open-source fused deposition modeling 3Dprinter (actually, we used the self-assembled cheap ver-sion Geeetech i3B).42 The used material was either anABS or a PLA filament; both resulted suitable enoughfor rigidness and durability. Designs were made using anopen-source software for both modeling (FreeCAD)43 andslicing (Slic3r)44. Printing was made in a not fully solidform, but with an infill of 40%, thus obtaining lighterparts almost equally strong. The files used for the print-ing will be available under a Creative Common license,together with the software code.34

A problem we had to face was that when printing inABS the part is prone to detach from the printer planeand to warp. This was a criticity where accurate orthog-onality or parallelism was required, namely in the webcamera and cell support and in the joints between mainand secondary beams. To overcome this problem critical

parts have been endowed with a simmetry where equalopposing deformations compensated. For this reason theweb camera is supported by twin parts mounted back-to-back, as described above. With this arrangement, thecamera axis resulted parallel to the main beam within±0.2◦, without further adjustment. A slight tilt in thelamp or in the opaline window support, instead, was nota problem.

Using ABS filaments with different colors we observedthat warping depended upon the color and was greaterwith the black filament. We could not determine defini-tively the reason for this behaviour, whether due to aslightly different chemical composition of the filament orto a different thermal behaviour due to the color. Weused a black or dark colored filament anyhow, to reducestray light in the optical system.

C. The dropping tip

The most critical experimental part in every ADSAapparatus is the tip that supports the drop. To obtaintrue axial symmetry, it is required that the tip orificebe perfectly circular, plane, adjusted to stand horizontal,and with a clearcut edge to avoid side attaching of thedrop. Moreover, all parts in contact with the drop liquidare to be made with a chemically inert material, to avoidany contamination by release of surfactant molecules.

Indeed, experience shows that the manufacturing of agood dropping tip is more of an art than a science andrequires some degree of expertise and craftsmanship (see,e.g., instructions in Ref. 45, pag. 51).

We tried several approaches. Some usable teflon tipswere manufactured in the Department mechanical facil-ity, by precision lathe machining. However, we did neversucceed in obtaining really smooth and clean tips thisway. Best results were obtained using an hypodermicplastic syringe without the needle, with the drop hang-ing from the adapter tip orifice (2.7 mm inner diameter).

The whole preparation procedure was moderately la-borious, did not require any specialized instrumentionand took only a few minutes. First, a syringe was in-spected visually with a magnifying glass; a syringe withan irregular or deformed tip was discarded. More often,the tip was regular, but there were attached some smallplastic fringes, due to the manufacturing process. Thefringes were removed by gently peeling the tip surfacewith a sharp razor blade. After this shaving, some tipswere already usable. As a further step, it was beneficial ashort annealing, exposing the tip to an air flow at about160 ◦C for 2 to 3 s. Both temperature and exposuretime were adjusted by trial to induce an initial surfacemelting that smoothed away the residual tip roughness,without any bulk deformation. To avoid side bending ofthe tip, the hot air gun must be directed with the air flowupward, keeping the syringe vertical. The tips manufac-tured with this procedure resulted of fairly good qualityand on average behaved better than teflon tips obtained

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by lathe machining.The tip was connected to the liquid reservoir either

using a 4 mm polyethylene tubing commonly used forcompressed air systems or a thin teflon tube 1 mm di-ameter. The polyethylene tube was tapered by soften-ing with a hot air flow while stretching, and forced intothe needle adapter from inside. Other tubings, made ofdifferent materials, were tried. With PVC, a clear pro-gressive lowering of surface tension was observed, of theorder of several mN/m over a time of a few tens of sec-onds. A similar effect was observed with silicon tubing,though to a much lower extent. Only polyethylene andteflon confirmed to be a clean enough material for thiskind of measurements.

All tests to validate the performance of the apparatushave been made using high purity water obtained witha Millipore Mill-Q system fed with low mineral contentspring water. The obvious reason for using water dropsis that water is by far the most studied liquid in all itsphysicochemical properties and highly accurate data arelargely available in the literature.

The drops where formed under computer control usinga piston burette (Metrohm Dosimat 665) to dispense aknown volume. However, a piston burette is a piece ofinstrumentation not easily available at a low cost. Alter-natively, drops of different size have been obtained usinga second upper syringe as a water reservoir, connectedto the drop supporting tip with a short 1 mm teflon tub-ing, fitted to the syringe needle adapters using pieces ofa micropipette tip cut as needed with a razor blade.

Using a Hoffman clamp to compress the teflon tubeand varying the reservoir vertical position, the flow couldbe adjusted for a drop time in the range from 5 to 100s. With a 60 s drop time, during the 30 ms electronicshutter time the volume change was about 0.005% of thedetachment volume, small enough to allow to considerthe drop practically static.

D. Software

For this project we have largely reused the softwarepackage described in Ref. 33 (and already made avail-able under an open source license). Hence, we refer tothat paper for the detailed description of the algorithms,whose operation has not been significantly modified bythe code updates described below. The main part of thatcode was composed of two libraries, one with the rou-tines for the extraction of the drop profile from an imageframe (profilib), the other with the routines for theleast-squares fit of the extracted profile with Bashforthand Adams equation, in order to obtain the drop apexcurvature and shape parameters R0 and β (fitlib).

The original code was written in Fortran, only partiallyported to C, and was designed to work only with a squareframe and the geometry of a sessile drop with the verticalaxis positive downwards. The last two characteristicswere of hindrance with the actual apparatus, because web

cameras usually have a rectangular frame, typically witha 4:3 aspect ratio. With a rectangular frame it is moreconvenient to use the landscape mode when working witha sessile drop, portrait mode with a hanging drop. Hence,routines in the profilib library have been modified towork with all four possible drop axis orientations among0◦, 90◦, 180◦ and 270◦. This solution has been consideredmore convenient than rotating the frame to adapt theimage to a predefined orientation, because of the largeframe size (about 5 MB).

In short, in the original algorithm there were indepen-dent routines, each designed to work on one among thetop, left, right and bottom sides of the profile (limitedby angles ϕ = ±45◦,±135◦), scanning from top of thesessile drop downwards, following the profile respectivelyclockwise (CW) along the right side and counterclockwise(CCW) along the left side. The original routines havebeen replaced by a single routine that can scan both CWand CCW along every side. We refer to the source codefor all other details.

The fitlib library algorithms instead were left intheir original form and the profile rotation, as well as thescan direction adjustment (CW), was performed onto thesmall vector containing the profile point coordinates (afew kB).

Both libraries have been fully ported to C, removingany remaining Fortran code, and two application pro-grams have been added. The first one (proget) is a frontend to the profilib library, the second (profit) to thefitlib library. Further details can be found togetherwith the source code.34

The procedure for the camera handling and the imageshot has been setup completely relying on the softwarepackages available with the Linux system for video device(v4l2).32 Hence, an image was taken with a short shellscript like:

#!\bin/bashv4l2-ctl -c auto-exposure=1 --set-fmt-video=

width=2592,height=1944,pixelformat=YU12

echo -e "P5\n2592 1944\n255" > image.pnm

dd if=/dev/video0 of=image.pnm oflag=appendconv=notrunc bs=5038848 count=1

In the first line the v4l2-ctl system program is used toset the camera parameters to the required operating con-ditions. Usually only a few parameters are to be changedwith respect to the system default values. In this case,the auto exposure system is turned off to keep exposi-tion independent from the drop size, the frame size isset to the maximum available resolution and a conve-nient pixel format is selected. The second line creates aproper heading for a PNM image file46 (image.pnm). Thethird line appends the frame to the head, using the an-cient and popular dd program to transfer data from the/dev/video0 camera device to the file. With the PNMformat, the image after the heading is raw, i.e. the file

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directly contains the array of 2592 × 1944 = 5038848bytes, each one containing the luminance of the corre-sponding pixel. The color information, that follows theluminance array, is useless for this experiment and is dis-carded simply not reading it.

This way the whole measuring process is composed ofthree completely independent steps. This arrangementis not very efficient from the point of view of speed. Asingle program in charge of driving the camera, taking theimage and using the profilib and fitlib libraries forextracting and fitting the drop profile, without copyingthe whole frame and the extracted point coordinates toa file, would be by far more efficient. However, this lastarrangement is best suited for a production system, forapplications of the ADSA technique in long repetitiveexperiments. From a didactical point of view, the usedapproach has the great advantage of making every stepindependent, repeatable and clearly understandable.

To simplify the system operation and (optionally) han-dle the whole measurement sequence without using thecommand line interface, a graphical interface has beenadded, written in Tcl/Tk (giada, Graphical Interface forAxisymmetric Dropshape Analysis).

The giada graphical interface window is shown inFig. 4. The camera is used in portrait mode, hence thedrop image appears rotated by 90◦. Four graphical ele-ments added to the drop image make it possible to set theworking conditions for the profile elaboration. A smallarrow (middle of the right edge in the image) fixes thestarting point and direction for going to intercept andfollow the drop profile. The two thin vertical lines at thebase of the drop mark the position of the syringe tip,where the profile ends, and the profile part to be dis-carded during the fit: points too close to the tip are notto be used in computations because they may be affectedby any minimal tip deformation. A respect band from20 to 50 pixel (' 0.1 to 0.25 mm) proved effective. Thesmall black and white boxes, middle of right edge nearthe arrow and near the tip edge respectively, determinetwo selected areas where to measure the dark and lightlevels.

All these elements can be modified as needed usingthe mouse. The bars above the image show the coordi-nates and value of the pixel pointed by the mouse, con-tain the command buttons to take an image, to extractand fit the drop profile and to navigate among the takenand archived images. A menu gives access to the sys-tem setup, that allows a fine trimming of all operations.Other details of the graphical interface can be found to-gether with the source code.34

All the software developed for this project and de-scribed in this paragraph, as already stated, is freelyavailable online under an open source licence. It canbe used at different levels of complexity. At the outerlevel, one can use the graphical interface and deal onlywith the measure of interfacial tension and the physico-chemical properties of the drop surface. Below the firstlayer, at the next level, the programs are available with

FIG. 4. Screenshot of the giada graphical interface for theADSA algorithms. See text or Ref. 34 for description.

the command-line interface and a much more detailedanalysis of the profile data can be performed. At a stilllower level, the intersted reader can find the source codeand interact with the inner algorithms used for the pro-file analysis and fit. The authors shall be grateful for anycommunication of errors found or code improvements.

E. Alignement, calibration and sources of error

ADSA is an absolute measure technique and no cali-bration is due with liquids with already known interfacialtension. Yet, the value of γ is obtained through eq. 7,where quantities g, D1 and D2 appear and are to beknown. Frequently these values can be found in the lit-erature or can be measured with a precision high enoughto make their error contributions negligible; these errorswill not be considered here.

The value of R0, the drop curvature in the apex (=r0/S), has to be obtained combining the value r0 in pixelobtained by the profile processing algorithm (togetherwith β) with the image scale factor S (pixel per lengthunity) of the optical assembly. Calibration of an ADSAapparatus means principally measuring this quantity.

Scale factor S has been obtained following a proceduresimilar to the one already described in Ref. 25, 29, and33. A calibration sphere was positioned at the tip orifice,and held in place by depression created with a Venturitube connected to the tip. A high precision 6 mm rubinball (from Comadur SA, Le Locle, CH) and a commercialstainless steel bearing ball of similar size, whose diame-ter was measured with a micrometer, gave same resultswithin 0.016% (equivalent to the resolution of the mi-crometer).

Error in the scale factor is due mostly to any differencein the distance between the meridian plane of the dropor the calibration sphere and the camera. Three con-

8

tributions have been considered: position reproducibilitywhen removing and replacing the cell cover to place orremove the calibration sphere; mechanical drift of theplastic supports; difference in the position of the merid-ian plane between sphere and drop because of syringe tiptilt.

The first contribution, measured repeating a few timesthe calibration procedure removing and replacing the cal-ibration sphere, resulted to be < 0.005%. The mechan-ical drift contribution, measured repeatedly taking theimage of the calibration sphere along one hour, resulted' 0.02% in the presence of a temperature drift of about0.5◦C. The third contribution was evaluated taking theimage of the calibration sphere after a syringe rotationof 0◦, 90◦, 180◦, 270◦ and 360◦. This gives an estimateof the effects due to deviation of the tip orifice from hor-izontality. Results in the range from 0.08% to 0.4% havebeen found, with most of the tested syringes giving avalue around ±0.15%.

The first two results show that the printed mechanicalparts have a good overall stability and reproducibility.The third result points to a problem in the horizontalityof the tip orifice, due both to the syringe, that is nota precision instrument, and to the cell cover, where thehole, where the syringe is fitted, may slightly deviate fromperpendicularity. Due to the square in Eq. 7, the thirdcontribution brings a typical ±0.3% error in the measureof γ. For water in ordinary conditions this amounts to±0.2 mN/m. Anyhow, these results were obtained withthe original short focal length optics, with a camera todrop distance of about 40 mm. Replacing the original op-tics with a lens with 8 cm focal length, all distances scaleup about one order of magnitude, reducing by the sameamount the errors discussed above. In this condition, themost important term drops to ±0.02 mN/m.

In the last, the most important source of error is inthe value of the shape factor β. Equation 7 has beenobtained with the assumption of several conditions:

1) the camera axis (the dashed-dotted line in Fig. 2) ishorizontal;

2) the frame Z axis is aligned along the local vertical;3) the drop image in the frame is a faithful copy of the

drop shape, with no geometrical distortion;4) the drop hangs from a perfectly circular flat and

horizontal orifice.Horizontal alignement (1) was accurate enough if the

upper beam of the apparatus was aligned horizontal witha spirit level, adjusting the supporting screws. The resid-ual divergence between the beam and the camera axis re-sulted to be negligible (< 0.2◦). Vertical alignement (2)was obtained observing a thin plumb line placed in themeridian plane of the drop, in the middle of the frame,and adjusting the supporting screws to make it alignedwith a pixel column. It has been shown in Ref. 33 thata vertical tilt well above the easily observable 1 pixel /1000 pixels has a negligible effect because of the profilesymmetry.

On the other side, condition (3), namely the absence

of geometrical distortion by the optical system, was verydifficult to be fulfilled using the original camera optics.At first we attempted to correct for the distortion, butthis procedure turned out to be unfeasible. The accuratemeasure of the correction parameters resulted awkwardand often the correction did increase the error instead ofreducing it. Moreover, the errors due to optical distortionmix inextricably with the effects of any irregularity in thetip orifice that affect the simmetry of the drop and alterthe apparent shape factor β (condition 4).

This problem, that might devoid of any validity themeasurement technique, has been overcome adoptinga measuring procedure that contains a self-consistencycheck of the obtained data.

It has been observed that the effect of both opticaldistortion and profile deformation due to tip defects isdramatic for the smaller drops (up to several mN/m)and becomes smaller and smaller as the drop grows. Forthe bigger drop, just before detachment, the effects caneasily be smaller than 0.1 mN/m.

Intuitively, it can be said that small drops have an al-most spherical shape, as stated by Young-Laplace equa-tion, and become more and more deformed because ofgravity as they grow, making less and less important thefurther deformation due to optics. On the other side,profile deformation due to tip defects rapidly decays withdistance from the tip. It is an everyday observation thatsquare taps do not create square drops. Whathever theshape of the tip profile, every drop rapidly gain a circularsimmetry with distance from the tip.

Hence, we adopted the following self-consistency check:if the r0

2/β value is measured while progressively increas-ing the drop size up to the detachment point and anasymptotically constant value is obtained for the largerdrops, this means that profile deformation due to dis-tortion (3) and (4) has become negligible in front of theintrinsic drop shape and the asymptotic value must bethe correct one

After the substitution of the original camera opticswith a lens, as described in paragraph III-A, effects ofoptical distortion and error on scale factor S dropped toa negligible level, leaving the quality of the tip orifice themain source of measure error.

IV. RESULTS

The self-consistency check described at the end of lastparagraph has been repeated testing tips manufacturedusing the different procedures described in Section III-C.During each test a set of images taken on a series of 20to 100 drops was analyzed. Every drop was formed dis-pensing a water volume slightly greater than the volumelost at detachment (about 50 µl), every time generatinga new fresh drop slightly greater than the previous one.After a delay from 4 to 10 s, to allow for the settlingof any surface vibration, 3 to 15 images were taken in asequence, with a cadence of about 2-3 s per image. In

9

some data set an appreciable drift in time of the β/R20

value in the image sequence on the same drop denotedthe presence of contamination. In that case the data setwas discarded (using the polyethylene tubing to connectthe water reservoir to the syringe tip, no appreciable driftcould be observed over a time not only of tens of secondsbut up to several tens of minutes as well).

The dispensed volume was so adjusted that each setcontained from two to five full series of drop images fromthe smallest to the greatest.

In every drop series, the value of r02/β was charac-

terized by an erratic behaviour when the drop volumewas small, typically less than 20 µl. When the erraticbehaviour persisted for drop volumes between 20 and 50µl, the tip was discarded. More often, above 20 µl ther0

2/β value converged to a constant, denoting a goodworking tip and yielding through Eq. 7 a reliable valuefor γ.

The results obtained using the original camera optics in25 independent test series are shown in Fig. 5. Each value

-1

-0.5

0

0.5

1

0 5 10 15 20 25

∆γ

(m

N/m

)

series number

FIG. 5. Differences among the measured values of interfacialtension of water and literature values. During measurementstemperature varied from 18.6 ◦C to 21.8 ◦C and was moni-tored with ±0.2 ◦ accuracy. This test was performed usingthe original camera optics.

reported in the plot has been computed averaging the re-sults in the asymptotic constant region and subtractingthe value from the literature at the corresponding tem-perature. For the comparison we have used the data from0 to 50 ◦ C in the International Tables of the surface Ten-sion of Water,47 interpolating with a quadratic functionto obtain values at the work temperature (temperature,in the range from 18 to 26 ◦C, was not controlled dur-ing the experiments, but only monitored with a ±0.2◦ Caccuracy).

The average difference for the whole series is −0.05mN/m (dashed line in the figure), with a standard de-viation of ±0.22 mN/m for the single value. This re-sult agrees very well with the error expected from thetwo most important contributions described in SectionIII-D, namely the effect of tip horizontal tilt (evalu-ated to ± 0.2 mN/m) and distortion effects (evaluatedto ± 0.1 mN/m).

Similar tests were performed after substitution of theoriginal optics with the lens, as described in section III-A.

The remarkable difference found with the new instru-mental arrangement is that differences computed withsame procedure of Fig. 5 were invariably very close tozero or negative. This finding completely agrees with thehypothesis that scattering of data in Fig. 5 is mostly dueto random errors in scale factor S, with equal chancefor positive and negative deviations. After reducing toan almost negligible value this error, the main source oferror in the tests became the presence of any trace ofcontaminants in the water, whose effect is always a low-ering of interfacial tension. As a fact, when a value ofγ was measured in defect with respect to literature andit was possible to repeat the test after loading fresh wa-ter and/or a thourough cleaning of tip and tubes, a newvalue in agreement better than 0.1 mN/m was found.

Two results taken from these tests are shown in Fig. 6.Below 20 µl the value of γ appears to vary wildly up

70

71

72

73

74

75

0 10 20 30 40 50

γ (

mN

/m)

µl

70

71

72

73

74

75

0 10 20 30 40 50

γ (

mN

/m)

µl

FIG. 6. Values of γ obtained in two measurement series afterreplacing the original camera optics with a 8 cm focal lengthlens. Temperature was 24.2 ◦C for the left series and 25.7 ◦Cfor the right one. The dashed horizontal line shows the valueobtained averaging the data in the volume interval 20 to 55µl. Average values are 72.16±0.09 mN/m for the left plot and71.90±0.14 mN/m for the right one. Corresponding literaturevalues are 72.11 and 71.88 mN/m respectively.

and down; above 20 µl it soon converges to a constantvalue. Averaging data for volumes greater than 20 µl,the difference from the literature value is 0.05 and 0.02mN/m respectively for the left and right plot, with rmsdeviation of the averaged values of 0.09 and 0.14 mN/m.

A last test to be reported has been made using thearrangement described at end of Section III-C to obtainimages of drops of different size without using the Dosi-mat piston burette. During a continuous water flow, asequence of 100 images was taken at a rate of about 3.3s per image with a drop time of 65 s, obtaining the lifehistory of about 5 different drops. Results of this ex-periment are shown in Fig. 7. Here also we have founda fairly good asymptotic behaviour in agreement withthe literature value (this test was made with the originalcamera optics).

The final remark is that despite the simple and echo-nomical resources used for the assemblage of this appa-ratus, the level of accuracy that can be obtained is fairlygood. The apparatus can be used even in its more sim-

10

70

71

72

73

74

75

0 10 20 30 40 50

γ (

mN

/m)

µl

FIG. 7. Values of γ obtained in a measurement series of100 images taken during the life of five slowly growing drops.Temperature during measurements was 26.2 ◦C.

ple arrangement, avoiding the use of any high level pieceof laboratory instrumentation, like the piston burette.Plastic syringes for sanitary use, though not very accu-rate mechanically, are very good from a chemical point ofview and can be easily adapted to work as a fair droppingtip.

V. ACKNOWLEDGMENTS

The authors are indebted with Dr. Giuseppe Molesinifor many helpful discussions about optics and for provid-ing the lens.

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