PSFVIP-8: The 8 th Pacfc Symposum on Flow Vsualzaton and Image Processng, Moscow, Russa, August st-5th, SURFACE TENSION AND DYNAMIC CONTACT ANGLE ANALYSIS FOR LIQUID DROPLETS Mzotn MM, Krylov A S, Gusev SA, Protsenko PV Lomonosov Moscow State Unversty, Faculty of Computatonal Mathematcs and Cybernetcs, Laboratory of Mathematcal Methods of Image Processng, Moscow, Russa Lomonosov Moscow State Unversty, Department of Chemstry, Moscow, Russa ABSTRACT The methods of the surface tenson measurement usng the pendant or sessle drop profle are well known for ther relablty and unversalty The method procedures are well establshed but stll requre hgh qualty drop mages to measure the surface tenson automatcally, what may be the problem n dffcult expermental condtons In ths paper, the hghly automated methods of surface tenson and dynamc contact angle measurement are presented The usage of mage processng technques allows processng of mages of dfferent qualty, acqured eg at hgh temperatures INTRODUCTION Drop shape methods are wdely used due to many advantages They are easy to handle, requre only small amount of the lqud and can be used n many dffcult expermental condtons such as hgh pressure, temperature etc The method can be used to measure the surface tenson of varous materals, from organc lquds to metallc melts and from pure solvents to concentrated solutons Also, the expermental procedure allows capturng the evoluton of dynamc systems to study tme dependent propertes The balance of gravty force and surface tenson forms the dstnct profle shape, whch can be descrbed usng the Young-Laplace equaton γ + P R R, ( where R and R are the two prncpal rad of curvature, and P s the pressure dfference across the nterface For a general rregular drop form, the ntegraton of the equaton ( s qute dffcult Fortunately, n the case of axally symmetrc drop, the equaton becomes twodmensonal, and effcent numercal procedures for ths case have been developed The nverse problem of the surface tenson determnaton from the drop shape s much more dffcult The frst efforts n ths problem were made by Bashforth and Adams [], who composed the tables of drop profles for dfferent surface tenson values and radus of curvature at the apex of the drop Untl the powerful computers became wdely avalable, the number of methods usng varous smplfcatons was proposed, but ther precson and range of applcablty was sgnfcantly lmted The major mprovement over the exstng methods was the ntroducton of the modern Axsymmetrc Drop Shape Profle (ADSA algorthm [] The Rotenberg s procedure fts the measured profle to a Laplacan curve usng a nonlnear procedure In ADSA, the objectve functon, whch s used to evaluate the dscrepancy between the theoretcal Laplacan curve and the actual profle, s the sum of the squares of the normal dstances between the measured ponts (e expermental curve and the calculated curve In addton, the locaton of the apex of the drop s assumed to be unknown and the coordnates of the orgn are consdered as ndependent varables of the objectve functon Thus, the drop shape can be measured from any convenent frame The second generaton of ADSA, developed by del Río [3, 4] overcame the defcences of the numercal schemes of the frst generaton usng more effcent algorthms He used the curvature at the apex rather than the radus of curvature and the angle of vertcal algnment as optmzaton parameters Today the numercal ADSA procedure s well establshed [5], wth only known lmtaton to near sphercal drops, where the algorthm hts the fundamental lmtatons of round-off errors and measurements precson Despte ths fact, the development of the method s stll not complete In contrast to the development of the numercal procedure, the mage acquston and profle extracton step was not thoroughly consdered Whle ths operaton may be trval n almost deal case of professonal laboratory equpment, n the case of dffcult condtons and nexpensve setup, profle extracton may be trcky For nstance, a more sophstcated mage analyss technque s requred for poor qualty mages of bubbles, eg n a dsperson, than for pendant or sessle drop mages n ar In ths paper, a couple of mprovements for the profle extracton step are proposed Also, the applcaton of the ADSA method to the quasstatonary drops s consdered and the method of contact angle determnaton s proposed NUMERICAL PROCEDURE In the case of axsymmetrc drop, the equaton for the drop profle shape ( can be reduced to the system of ordnary dfferental equatons y x c( y y + + b ; x x x c( y y + + b, x x wth ntal condtons y ( y, x ( x, x (, ( and also, b Here, b s the curvature at x t ρg the drop apex and c s the capllary constant ( ρ γ (
PSFVIP-8: The 8 th Pacfc Symposum on Flow Vsualzaton and Image Processng, Moscow, Russa, August st-5th, s the densty dfference between drop and the medum, g s the gravty acceleraton To determne the surface tenson t s needed to solve the nverse problem for the system ( to fnd the parameters, y, c, b} In practce, t s also needed to determne the camera tlt angle The soluton s performed va mnmzaton of the total error E ( x, y },( d {, (3 where d(, y },( s the dstance between, y and the calculated lne expermental pont } ( (see fg PRECISION EVALUATION To estmate the precson of the developed method of surface tenson calculaton the numercal experment was held The theoretcal profles wth parameter / b R 4, and dfferent capllary constants c 3 7 и c were generated The method performance was evaluated for dfferent levels of the total error δ δ N ~ ρ ( X, X, where X are coordnates of the theoretcal profle ponts, X ~ are coordnates of the perturbed drop profle ponts, ρ s dstance between ponts n drecton of the normal to the theoretc drop profle (4 For the experment, ponts were taken The perturbed values X ~ were calculated by addng unformly dstrbuted nose to the ponts X In the Table the results from runs for each nose level are presented Table Precson evaluaton of the capllary constant determnaton by the Laplace method Fg Theoretcal drop shape (sold lne and expermental ponts (dots Thus we need to calculate the dstance between expermental ponts and the soluton of ( The system ( does not have analytcal soluton, and t s solved numercally So the dstance d ({ x, y },( mn( x t x( t + ( y t has to be found by the mnmzaton procedure Due to effcency reasons, the followng procedure s performed The approxmaton error s evaluated smultaneously wth the ntegraton of ( The expermental ponts, y} are sorted by the mage processng algorthm such that for each next pont the correspondng arc length parameter t s ncreasng Then, n the process of ntegraton the last value of t can be used as ntal estmate for t + Durng the ntegraton va Dormand-Prnce method [6] at each step the values ( x ( ( are avalable, thus the t can be determned from the condton ( x x( t x + ( y t (5 usng Newton method It s only needed to track the value of the left sde of (5 durng the ntegraton (4 R 4 c 3 7 c δ c mn c rel c mn c rel 379 379 9 9993 9993 7 3665 37779 9954 9 46 369 37933 5 99 73 78 3 35664 3846 383 9883 7 4 3539 38899 53 9845 47 55 5 3483 39399 648 987 84 93 6 3447 399 787 9769 3 3847 447 39 96 375 38 The values c mn и c represents mnmum and mum recovered values of c for each value of δ, also the table gves the relatve errors of c determnaton: { c c, c c } mn rel c The examples of the determned drop parameters and drop profles are gven n fgs, 3 The perturbed values of ponts X ~ are denoted wth small crcles and the recovered profle s denoted as sold lne In both cases δ was taken The exact solutons are not gven because they are vsually matched almost exactly
PSFVIP-8: The 8 th Pacfc Symposum on Flow Vsualzaton and Image Processng, Moscow, Russa, August st-5th, Fg The recovered profle for the parameter values R 4, c 37, δ Fg 4 Gonometer LK- The desgned software ncludes unversal vdeo capture module workng wth any hardware supportng DrectShow vdeo capture standard (almost any dgtal camera compatble wth Mcrosoft Wndows Ths enables us to use wdely avalable nexpensve dgtal cameras wth dfferent resoluton; the adjustment to the hardware s performed va specal calbraton procedure Fg 3 The recovered profle for the parameter values R 4, c, δ EXPERIMENTAL SETUPS Two expermental setups were used n ths study to test the effcency of developed algorthm The gonometer LK- (OpenScence Ltd conssted essentally from horzontal mcroscope equpped wth 4 x / objectve, USB CCD camera (8x4, up to 5 fps and mechancal substage for sample postonng (fg Drop of the lqud under nvestgaton s deposted manually by means of the syrnge on selected substrate surface, than the drop s postoned to the optcal axs of the mcroscope and the system s focused Indvdual mages or mage seres are transferred to PC Specal equpment was buld up to study fast spreadng of low vscosty alkal halde melts over ceramc substrates n ar Substrate s placed nsde tubular furnace and heated up to desred expermental temperature Substance to be melted s placed nsde alumna crucble whch s stuated over substrate When expermental temperature s reached, melt s extruded from the crucble through the capllary made of hexagonal BN and deposted on substrate Durng deposton sequence of drop mages s regstered wth specal dgtal camera FastVdeo 4 (4 fps at 64x48 and stored on PC Software package The software package s desgned to support the measurements of surface tenson and contact angles n a wde class of condtons wth a hgh amount of automaton Fg 5 An example of drop seres processng developed program The software supports automated capture of frame seres to study systems wth dynamc parameters (as shown n fg 5 For the case of rapdly changng systems, where the Young Laplace equaton s not applcable, a specal method of automated contact angle measurement can be used Image processng As shown n [5], the most robust method of drop profle extracton s the Canny edge detecton algorthm [7] However, n dffcult condtons even the Canny method extracts spurous edges due to lghtng condtons, mage nose and presence of foregn objects n the mage To automate the process of drop profle extracton n such condtons the followng procedure s used Frst, the bnary edge map acqured by edge detector s organzed nto lsts of successve ponts usng edge lnkng algorthm [8], based on the analyss of the pxel neghborhood Ths stage s also essental for effcent error functon (3 evaluaton, as mentoned above After ths procedure, the
PSFVIP-8: The 8 th Pacfc Symposum on Flow Vsualzaton and Image Processng, Moscow, Russa, August st-5th, drop contour s usually just the longest one, and can be found automatcally In rare cases drop contour can be edted manually to jon broken contours or to remove obstacles To further refne the drop profle, the extenson of the Canny method s appled, provdng sub-pxel precson [9] INTERFACIAL TENSION AND CONTACT ANGLE MEASUREMENT The surface tenson measurement was tested n applcaton to statonary and quas-statonary drops Interfacal tenson measurement for quas-statonary drops n lysozyme water soluton / octane system The developed algorthm has been appled to measure nterfacal tenson between water soluton of lysozyme ( 4 M and octane Polytetrafluoroethylene (PTFE substrate was placed on the bottom of optcal glass cuvette Than the cuvette was flled wth octane and placed on substage of LK- gonometer The drop of lysozyme water soluton was placed on PTFE surface by means of a sampler (fg 6 A sequence of drop mages was recorded and nterfacal tenson durng 4 seconds was evaluated (fg 7 Durng ~5 s sharp decrease of nterfacal tenson to about 8 mn/m was observed Gradual decrease of nterfacal tenson from 8 to 4 mn/m durng next 35 s could be attrbuted to rearrangement of proten molecules n nterfacal layer Dynamc contact angle measurement for fast spreadng drops (lqud NaCl / hydroxyapatte (HAP system Low vscosty melts spreads fast over wettable substrates (for mllmetrc sze drops trple lne velocty up to m/s could be observed [] Fast CCD recorders should be used to reveal detals of spreadng process Although drop profle dd not satsfy the Young-Laplace equaton at such hgh spreadng rates, t s mportant to determne dynamc value of contact angle θ (t locally at trple lne regon Ths allows to estmate drvng force of spreadng (per unt length of trple lne: ( cosθ ( cos F σ lg t θ eq Thus, the contact angle has to be measured usng dfferent technques In most cases drop profle remans qute smooth even when the drop s spreadng rapdly Moreover, when the contact angle s small, the drop s near sphercal at the trple lne regon Consderng these facts we propose the followng algorthm for measurement of the spreadng drop contact angle Frst, the drop profle s extracted usng the same procedure as for statc and quas-statc drops e Canny method wth sub-pxel precson It should be noted here, that the sub-pxel precson s necessary for ths task, because of small resultng angles at the end of the spreadng process Second, the drop left and rght contact ponts are detected va the frame dfferencng Fnally, the angle s evaluated usng teratve fttng Because the drop s assumed to be symmetrcal, two parts from the both sdes of the drop profle are taken and ftted by a crcle For the begnnng, 5% of the drop profle s used (see fg 8, and ths percent teratvely ncreases whle the ft remans good, that s the mean approxmaton error does not exceed, pxel (ths s a precson of the used sub-pxel detecton algorthm, accordng to [9] Fg 6 Water drop on PTFE substrate mmersed n octane Fg 8 Dynamc contact angle measurement scheme The example of the ftted crcle for molten NaCl drop spreadng over hydroxyapatte at 866 C s presented n fg 9 The measured contact angles are gven n fg Fg 7 Interfacal tenson of octane/lysozyme water soluton system
PSFVIP-8: The 8 th Pacfc Symposum on Flow Vsualzaton and Image Processng, Moscow, Russa, August st-5th, мм Fg 9 Contact angle measurement of the spreadng drop of molten NaCl [4] del Río OI, Neumann AW J Collod Interface Sc vol 96 (997, 36 [5] M Hoorfar and A W Neumann, Advances n Collod and Interface Scence, vol (6, Issues -3, 3 pp 5-49 [6] Dormand, J R; Prnce, P J, Journal of Computatonal and Appled Mathematcs 6 (: 9 6 (98 [7] Canny J IEEE Trans Pattern Anal Mach Intell vol 8 (986, 679 [8] DY Chan, WC Wang Robust Edge-Based Object Segmentaton by Lkely Boundary Anchor Selectng and Adaptve-Thresholdng Lnkng Proc of the Conference on Computer Vson, Image Processng and Informaton Technology, Chng Yun Unversty, Zhongl, Jun 9, [9] Devernay F A non-ma suppresson method for edge detecton wth sub-pxel accuracy Techncal report RR 74, INRIA (995 [] Eustathopoulos N, Ncholas M, Drevet B Wettablty at hgh temperature, Pergamon materals seres: vol 3 Oxford, UK: Pergamon, 999 Fg Tme dependence of contact angle for NaCl drop spreadng over hydroxyapatte substrate CONCLUSION In ths work, the hghly automated methods of surface tenson and dynamc contact angle measurement are presented The usage of advanced mage processng technques allows processng of mages of dfferent qualty, acqured eg at hgh temperature and hgh pressure, n dspersons, etc The developed software package allows usng nexpensve expermental setups for express measurements of surface tenson and contact angles The numercal and practcal experments revealed the hgh relablty and good precson of the developed methods ACKNOWLENDGEMENTS The work was supported by federal target program Scentfc and scentfc-pedagogcal personnel of nnovatve Russa n 9-3 and by RFBR grant - 8-44-а REFERENCES [] Bashforth F, Adams JC An attempt to test the theory of capllary acton, Cambrdge (89 [] Rotenberg Y, Boruvka L, Neumann AW, J Collod Interface Sc, vol 93 (983, p 69 [3] del Río O I On the Generalzaton of Axsymmetrc Drop Shape Analyss, MASc Thess, Unversty of Toronto, Toronto, 993