kinematic characterization of the advancing liquid–gas interface in microfluidic components...

Experimental Thermal and Fluid Science 38 (2012) 40–53

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Experimental Thermal and Fluid Science

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Kinematic characterization of the advancing liquid–gas interface in microfluidiccomponents combining digital processing of microscope images with curvematching technique

Nuno Rocha, Viriato Semiao ⇑TU Lisbon, Inst. Super. Tecn., IDMEC, Dept. Mech. Eng., Av. Rovisco Pais, P-1049001 Lisbon, Portugal

a r t i c l e i n f o

Article history:Received 27 July 2011Received in revised form 7 November 2011Accepted 13 November 2011Available online 20 November 2011

Keywords:Microflow visualizationDigital image processingCurve matchingUnsteady two-phase microflowsInterface advancing velocities

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⇑ Corresponding author. Tel.: +351 218417726; faxE-mail address: [email protected] (V. Semia

a b s t r a c t

A method based on flow visualization to measure the advancing planar liquid–gas interface velocities ofunsteady two-phase microflows is presented. A high-frequency CCD camera connected to a microscope isused to acquire flow images, which are digitally processed to define mathematically the interface loca-tions at each frame of the acquired images with accuracy of ±½ pixel. Curve matching technique includ-ing a function to define the optimal matching path is applied to the parameterized interface locationspreviously obtained to determine the advancing interface velocities. The continuity equation is includedin the algorithm to evaluate the velocity profile that minimizes the mass imbalance. The method is appro-priate for open curves typical of moving interfaces inside confined microdevices and is applied to thefilling process flows at constant rate, imposed by a syringe pump, in two constant height microchannelswith different sizes for validation purposes. The results yielded by the methodology prove that it isaccurate in determining planar flows with no vertical velocity component with mass balance errorsbelow 0.07%. When a relevant vertical velocity component is present, the technique is hardly applicableto such flows since it is unable to capture the interface movement in that direction, yielding errors ofmass balance that can go up to 41.6% in the case of a triangular microvalve with varying height.

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1. Introduction the different device components so that the physical phenomena

Microfluidic systems manipulate very small fluid volumes(fL–mL) flowing through their components with characteristiclengths in the range 1–1000 lm. Two novel concepts of microfluidicsystems were introduced in the 1990s as devices to performextracorporeal laboratory assays, such as blood analyses [1]: theLab-on-a-Chip and the Lab-on-a-CD. To carry out such tasks bothdevices contain components designed for specific functions likemicrochannels, microvalves, flow-splitters, mixers, micro-reservoirsand micro-sensors, but use different driving forces, the Lab-on-a-CDbeing driven by the centrifugal force originated from the diskrotation.

As described by Oh and Ahn [2], such new laboratorial microsys-tems allow for: (i) the enhancement of transport phenomena due tothe increase of surface/volume ratio; (ii) the drastic time reductionfor chemical reactions, and; (iii) the much smaller consumption ofpotentially expensive reagents and samples. Even though, thepresent stage of development of such microtechnologies requiresfurther research for a systematic characterization of the flows in

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governing their operation become enlightened [2].The growing interest in microfluidic applications led to an

intensive research of the filling process of microcomponents. Inspite of that, the kinematic characterization of such process is stillmaturing and techniques capable of quantifying and identifyingthe mechanisms governing such flows have to be developed andimproved. In fact, and regardless of the considerable number ofworks on the filling process of microcomponents, e.g., [3–12], theresulting characterization has been limited to simple cases.

Even for the simplest geometry, the straight microchannels,there is a lack of knowledge on the physics of their filling processes.Based on the pioneer theoretical work of Washburn [3] on capillaryflows, several authors (e.g., [4–6]) investigated experimentally thefilling process of microchannels by using one-dimensional ap-proaches (like the Hagen-Poiseuille fully developed flow solution)to calculate the spatial-average velocity of the liquid–gas interfaceadvance. However, the liquid–gas interface exhibits a velocity pro-file that varies in space and time. Moreover, the filling processes ofother devices like microvalves and flow-splitters are quite diversefrom that in a microchannel, being far from a well-behaved fullydeveloped flow and, therefore, they require more complex andsuitable approaches.

Splitters should provide an even flow splitting [10] in enzyme-linked immunosorbent assays (ELISA). Junctions have also been

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Fig. 1. Scheme of the experimental set up. 1 – syringe pump; 2 – microchips; 3 –illumination system; 4 – microscope lens; 5 – incident light beam; 6 – reflectedlight beam; 7 – CCD camera; 8 – computer; 9 – optical inverted microscope.

N. Rocha, V. Semiao / Experimental Thermal and Fluid Science 38 (2012) 40–53 41

studied as mixing promoters, mixing being a difficult task toaccomplish in Lab-on-a-chip or Lab-on-a-CD applications due tothe markedly laminar flow character [11,12].

Other authors (e.g., [7–9]) studied experimental and numeri-cally the filling process of microvalves dedicated to regulate theflow, operating based on the decrease of the meniscus contact an-gle at a sudden expansion of the cross-section. However, most ofthose studies report only visualization experiments focused onthe validation of the models that predict the meniscus contact an-gle variation of the advancing liquid–gas interface but the kine-matic characterization of such devices is still to be done.

A considerably complex three-dimensional model for microval-ves applicable to large values of the ratio width/height of the cross-section, considering the interface as a hemisphere, was put forward[13]. In turn, Glière and Delattre [7] proposed an improved three-dimensional model, valid for any cross-section shape and for bothhorizontal and vertical expansions, but its complexity requires theuse of numerical methods. Other authors [8,14] used the previousmodel [7] applying it to all height/width ratio values of thecross-section and performed its validation using flow visualizationwith image processing, particularly the contrast enhancement orfiltering.

Despite the various works on the subject, improvement of phys-ical understanding and subsequent mathematical modeling of thefilling processes of components of microdevices still prevails. Forthat, new and improved characterizing techniques are required.

Image processing technique, which has been used in manydevelopment works, is quite standard. The application of this tech-nique started some 40 years ago and has been improved since thenfor applications to computer vision problems [15], microfluidics[16,17] and macro-scale fluid mechanics [18–20]. Such techniqueallows defining the liquid–gas interface location of a moving fluidat several distinct moments.

The knowledge of such positions may, in turn, be used to deter-mine the advancing velocity of that interface recurring to the curvematching technique, which has, so far, been only applied to recog-nition of speech and handwriting [21,22], artificial vision [23] andtracking of meteorological structures [24]. Such technique is char-acterized by quantifying the dissimilarities between two curvesusing appropriate functions. These dissimilarities can be relatedto the curvature [21], for the cases of small curves deformations,or to the geodesic distance maps of the surface defined by thecurves [24,25], in this case applicable only to closed curves.

As discussed above, and to the authors’ knowledge, it has neverbeen reported in the literature the combination of the digital imageprocessing and the curve matching techniques to evaluate theadvancing velocity of the open curve defining the liquid–gas inter-face in transient two-phase flows at microscale. In this work, a newmethod based on the curve matching technique including the con-tinuity equation to ensure the mass balance is presented and usedto quantify the transient flow velocities of the advancing liquid–gas interface in a horizontal plane during the filling process ofmicrocomponents under the action of an imposed volumetric flowrate. It should be noted that the definition of the velocity profileposition is based on an optimal path search, which is a novelty.

This method is only applicable to open curves typical of theliquid–gas moving interfaces in confined flows. These curves corre-spond to positions of the referred interface extracted by digitallyprocessing the visualization images acquired with a CCD (chargedcoupled device) camera connected to a microscope.

This new method is validated herein by applying it to the char-acterization of the kinematics of unsteady two-phase flows typicalof moving gas/liquid interfaces during the filling process of rectan-gular microchannels of constant height under an imposed flowrate. Moreover, and to demonstrate its limitations, the method isalso applied to the filling process of a triangular microvalve of

varying height (with a vertical expansion perpendicular to theview plane) that induces a vertical velocity component. The pres-ence of such velocity component hinders considerably the applica-tion of this method to geometries of varying height.

2. Experimental setup

A scheme of the experimental setup used in the present work isdisplayed in Fig. 1.

The illumination system consists of a 55 W tungsten lamp.A Phantom V4.2 CCD camera with a 22 lm pixel pitch in both

directions is used. For the maximum resolution used, 512 pix-els � 512 pixels, corresponding to 11,264 lm � 11,264 lm, suchcamera allows for a maximum acquisition frequency of 2100 Hz.The total acquisition time for the defined acquisition frequencyand spatial resolution is limited by the 1 Gb memory of the CCDcamera and, therefore, each experiment has to be performed witha predefined acquisition frequency to ensure the recording of thecomplete filling process.

To guarantee the field of view needed for flow visualization, thecamera is connected to an optical inverted Leica� microscope,model DM ILM, with a lens possessing a magnification of 5� thatcorresponds to a numerical aperture (NA) of 0.12 and has adepth-of-field of 76,9 lm. Since the recorded images are amplifiedthrough the lens use, the spatial resolution of the object plane dif-fers from that of the CCD camera images. With the used lens, theCCD camera resolution (512 � 512 pixels � pixels) and the pixelpitch (22 lm) it is straightforward to obtain the object plane sizeof 2252.8 � 2252.8 lm � lm.

A New Era Pump Systems� syringe pump model NE-1000, cali-brated prior to the experiments, is used to ensure the desired con-stant volumetric flow rate.

The drawings displaying the schematics of the studied microflu-idic components are shown in Fig. 2. It should be noted that theseschemes do not depict the actual details of the geometries, as arethe cases of the surface irregularities due to the manufacture

Fig. 2. Scheme of the microcomponents manufactured by Biosurfit. (A) Rectangular microchannel of constant height and 217 lm of width. (B) Rectangular microchannel ofconstant height and 668 lm of width. (C) Rectangular microvalve with vertical expansion of height. (D) Triangular microvalve with vertical expansion of height. Formicrovalves (C) and (D) the vertical expansions are represented in a lighter gray color.

42 N. Rocha, V. Semiao / Experimental Thermal and Fluid Science 38 (2012) 40–53

process. The chips of the type shown in Fig. 1 containing thesegeometries were manufactured by Biosurfit.

Images (A and B) of Fig. 2 show the straight microchannels withconstant height that are used to validate the technique. Althoughthe manufacturer expressed the channels and valves heights(in the zz direction) for every chip, which are always above700 lm, the accurate value of such height is not a relevant param-eter for the experimental technique used herein. Therefore, such

Fig. 3. Schemes of the algorithms to determine the advancing velocity of the gas–liqumathematically the interface location; (B) Curve matching algorithm to determine the p

values are not expressed in Fig. 2, since: (i) this technique is onlyapplicable to flows without vertical velocity component and, there-fore, not applicable to valves expanding vertically, as it will bedemonstrated later on; (ii) the images recorded by the CCD camerarefer to the horizontal mid-channel planes with a depth-of-field of76.9 lm, which is one order of magnitude smaller than the chan-nels height (always above 700 lm). Therefore, the errors emergingfrom the interface projection onto the image plane are minimized.

id interface inside microdevices. (A) Digital image processing algorithm to definelanar advancing velocity profiles of the interface.


Images (C and D) of Fig. 2 depict the microvalves, respectively arectangular and a triangular shaped one, used to describe the tech-niques: the rectangular microvalve is used to describe the digitalimage processing technique that defines the liquid–gas interfaceposition, whereas the triangular one is used to describe the curvematching technique to calculate the interface advancing velocity.It should be mentioned that, besides the horizontal expansion (inthe xy plane), both microvalves exhibit a vertical expansion (inthe zz direction). The triangular microvalve is also used in thiswork to demonstrate the limitations of the technique.

Since the technique used is independent of the flow pattern(providing there is not a third velocity component) and of the usedfluid, tests were performed using as working fluid deionised waterkept at 21 ± 0.5 �C during the experiments.

3. Image analysis techniques

In order to quantify the marching velocity of the liquid–gasinterface inside microcomponents two different modelling algo-rithms were developed and used. Fig. 3 shows a scheme of both

Fig. 4. Sequence of selected images to illustrate the digital image processing algorithm(C) Region external to the microvalve: img_define_walls. (D) Liquid image: img_binary_position image: img_preliminary_interface_position. (G) Image with the detail of the moface_position. (H) Image with the detail of the same most right interface position after s

algorithms layouts and structures, their functions, sub-functionsand main purposes, so that a global overview of the entire proce-dure is obtained. The first algorithm, Fig. 3A, is dedicated to definemathematically the liquid–gas interface location and is based onflow visualization and processing of digital images acquired withthe CCD camera and microscope. The other algorithm, Fig. 3B,quantifies the planar advancing velocity profiles of the liquid–gasinterface by making recourse to the results yielded by the firstalgorithm and to the curve matching technique. Both algorithmsare described below in detail.

3.1. Algorithm to define the interface location of the advancing liquid–gas

For the sake of intelligibility the description of the algorithmused herein to extract the position of the advancing liquid–gasinterface from the acquired images of the filling process visualiza-tion and to define it mathematically is performed based on the ac-tual flow in the rectangular microvalve shown in Fig. 2C and on itsimages displayed in Fig. 4.

. (A) Arbitrary flow image: img_original. (B) Empty valve image: img_background.liquid. (E) Expanded air image: img_binary_air_expanded. (F) Preliminary interfacest right position of the final interface position before smoothing: img_final_inter-

moothing through polynomial fitting: img_smoothed_interface_position.


The flow image img_original sketched in Fig. 4A is an arbi-trary one. The empty valve image, img_background, establishingthe background needed for the algorithm, is displayed inFig. 4B. It should be mentioned that the smudge appearing inthe lower left-corner of the valve is a chip imperfection withno interference in the algorithm since it is located outside thevalve.

The different sequential functions, their objectives and opera-tions, the images they operate on and the yielded results are quite

Table 1AAlgorithm to define the liquid–gas interface location (starting images: img_original (4A) a

Function objective Function operations

Background_removal (i) Operates a contrast enhancemdefining the image by a constantimg_background (4B)

Establishes digitally the region external to themicrovalve starting from image 4(B)

(ii) Performs a threshold (gives aa gray-scale value below the estabimage(iii) Splits the last image into twright wall(iv) Converts the resulting imagemicrovalve walls location (all maassigned with the unity value; alunaffected)(v) Fills in the regions outside thetwo wall images(vi) Adds these binary-filled wallmatrix) and multiplies the outcom

Define_Interface_Location (i) Subtracts moderately the com4(A) and img_background 4(B) to

Locates the interface position starting fromimages 4(A) and 4(B)

(ii) Performs a contrast enhancem(iii) Performs a threshold operatiliquid region output – image 4(D(iv) Obtains the air image from thto its conversion to binary forma(v) Subtracts the result from the ioutcome of function Bakground_(vi) Operates a threshold and a c(vii) Applies a Gaussian filter (filtstandard deviation above 1 pixelexpand its external limits (img_bpreventing the intersection of thea void set of points (img_prelimin

a The non-void intersection establishes a first approximation of the interface locationnumber of identified interfaces is then evaluated and each interface is validated – see imand, for each valid interface, a local rectangle with the exterior positions of the interfac

Table 1BAlgorithm to refine the preliminary liquid–gas interface location (starting image: img_pre

Sub-function objective Function operations

Refine_Interface_Location (i) Validates the interface(number of pixels)

Evaluates the vector positions of an interface starting fromimage 4(F): called for each identified interface

(ii) Calculates the most dnarrow band since the aexpanded – image 4(G)a

(iii) Evaluates the interfaas a decision criterion fothat may become useful

Smooth_Interface (i) Splits the interface locadvancing from the mos

Performs a polynomial curve fitting startingfrom image 4(G)

(ii) Performs a normaliza[�1,1] by means of Legeorthogonal space of funcapply the least square m(iii) Performs a polynominterface location yieldinimage 4(H)

a The result is a stepwise shape of the interface location displayed in image 4(G), whicprocess, the pixel.

b There is a need for a referential rotation whenever a large interface location gradien

standard and summarized in Tables 1A and 1B that make recourseto Fig. 4(A–H). Matlab� with its standard functions was the soft-ware used to perform all the digital image processing operations,considering that each image is a set of pixels, with each pixelcorresponding to a value in the range [0–255] constituting anumerical matrix. Table 1A contains information on the operationsto define a preliminary interface location whereas Table 1B refersto the refinement of the definition of that location with accuracyof ±0.5 pixel (±2.2 lm) – see Fig. 4H.

nd img_background (4B)).

Resulting image

ent (product of the numerical matrix) to the complement image of

nil value to all the matrix elements withlished limit) on the previously obtained

o images to separate the left from the

s into binary format identifying thetrix elements with non-zero values arel matrix elements with zero values are

microvalve with a binary format for the

images into a single image (numericale by a factor of 255 yielding image 4(C)

Region external to the microvalveimg_define_walls 4(C)

plement images of both img_originalobtain the liquid imageent to the result Liquid region

on for binary conversion yielding the)

img_binary_liquid 4(D)

e complement of the liquid image priortmage img_define_walls (image 4 (C), theRemoval)onversion to the binary formater with size of 5 � 5 pixel � pixel and a) to the air binary image in order toinary_air_expanded – image 4(E)),liquid and air binary images from beingary_interface_position – image 4(F))a

Air region (expanded)img_binary_air_expanded 4(E)preliminary interface positionimg_preliminary_interface_position4(F)

that is constituted by a narrow band with a width that can go up to 4 pixels. Theage 4(F) where false interfaces appearing inside the marking circle are eliminated –e is determined and oversized.

liminary_interface_position 4(F)).

Resulting image

elements based on their area values

ownstream interface location in their image was the one previously

ce injectivity referring to x–directionr the need of a referential rotationfor function Smooth_Interfaceb

Non-interpolated final position ofinterface img_final_interface_position 4(G)

ation into organized vector positionst left to the most right interfacetion in the x-values in the intervalndre polynomials, which form antions in that interval as the basis toethod for the curve fittingial curve fitting to smooth theg img_final_interface_position –

Final position of interfaceimg_final_interface_position 4(H)

h is due to the inherent limitations of the smallest unit allowed by the image digital

t in the x–direction occurs.

Fig. 5. Sequence of selected images to illustrate the advancing velocity profiles calculation algorithm. (A) The first interface location C1 (x1, y1), with identification of thearbitrary section downstream C2 (x2, y2), the control surface CS and control line CL. (B) Two different interface locations, C1 (x1, y1) and C2 (x2, y2), separated by Dt = 16.67 ms.(C) Identification of the optimal path in the parameterized domain [C1(t), C2(s)]. (D) Displacement vectors. (E) Variation of the error as a function of k defining the iterativeprocedure to obtain the optimal path c.


Fig. 6. Selected images of the flow visualization of the filling process at constantflow rate of 3 ll/min, imposed by a syringe pump, in the rectangular microchannelwith a width of 217 lm, acquired with the CCD camera. (A) View of the emptymicrochannel; (B) Liquid–gas interface position at t = 0 s; (C) Liquid–gas interfaceposition at t = 0.05 s; (D) Liquid–gas interface position at t = 0.0833 s.


The illumination system used in the experiments has AC oscil-latory character and this might influence the contrast and bright-ness of the acquired images. Since the described algorithmobtains the liquid region by subtracting a defined frame with adevice with liquid – Fig. 4A – from the equivalent frame with thedevice empty – Fig. 4B – the signal-to-noise ratio decreases asthe contrast and brightness difference between those two imagesbecomes larger. To avoid this, the acquisition frequency used wasalways far superior to 50 Hz and the frames used to determinethe interface location were chosen ensuring similar contrast andbrightness.

Additionally, one can observe non-continuous liquid regionsclose to the walls, like those identified by the rectangles inFig. 4D. Their existence poses a difficulty in evaluating the interfacelocation at those locations since there is a band near the walls thathas a grayscale color quite close to the liquid one – see Fig. 4(B) –yielding a very low signal-to-noise ratio. Because of that, in thosenear wall bands the algorithm presented hardly finds points forthe interface definition.

3.2. Algorithm to evaluate the velocity of the advancing liquid–gasprofiles

Similarly to the previous description, the algorithm to obtainthe advancing velocities of the liquid–gas interface is alsodescribed based on an actual flow (see Fig. 5) in the triangularmicrovalve shown in Fig. 2D.

This algorithm uses that of Basri and Frenkel [21] describedbelow that matches any two parameterized open curves in a plane(according to their arc length) through their geometric proprietiesas an optimization problem. The optimal path in the domain de-fined by the arc-lengths parameterizations of two curves is chosenas the one that minimizes a specific cost function that relates thecurves geometric properties [21]. The result of this algorithm isthe correspondence on the discretized space between the twocurves.

Since the present work intends to quantify the advancing veloc-ities of a liquid–gas interface, the curves we have used are thosethat express such interface location in different images acquiredwith the CCD camera followed by the digital image processing asdescribed in Section 3.1, and their correspondence is made interms of displacement vectors. As the time interval between theacquisitions of two images is known, the velocity vector is easilyobtained by dividing each displacement vector by the mentionedtime interval.

Two new functions are introduced in the present work: one todefine the velocity profile based on an optimal path search andthe other to evaluate the error though the mass imbalance.

The algorithm is composed of several functions as describedbelow in detail and, for any two given curves C1 and C2, it intendsto determine the necessary deformation to obtain C2, startingfrom C1.

3.2.1. The Basri and Frenkel [21] algorithmConsider two curves defining the liquid–gas interface locations

acquired with the CCD camera after the digitally processing of theimages with the algorithm described in Section 3.1: one named asfirst interface position in Fig. 5A and represented as C1(x1, y1) inFig. 5B and another downstream the previous one, obtained aftera time interval of say Dt = 16.67 ms, represented as C2(x2, y2) inFig. 5B.

The Basri and Frenkel [21] algorithm comprises the functionsdescribed below – refer to Fig. 3B.

3.2.1.1. Function curvature_calculation. This function parameterizesboth interface locations C1(x1, y1) and C2(x2, y2) as a function of

their lengths calculated from their arc-lengths dt and ds – seeFig. 5B – and determines their curvatures.

For that, the curves representing the mentioned interface loca-tions are first discretized into a sufficiently large number of gridnodes: m for curve C1(x1, y1) and n for curve C2(x2, y2), where xand y are the coordinates in the physical space. The number of gridnodes m and n must be chosen so that the obtained results aregrid-independent.

The linear distance between any two consecutive nodes of thesame curve that measures approximately the arc length, dt forcurve C1(x1, y1) and ds for curve C2(x2, y2), is then calculated asdt ¼ ðdx2

1 þ dy21Þ

1=2 or ds ¼ ðdx22 þ dy2

2Þ1=2, where dx and dy are the

discretized horizontal and vertical distances between two neighborgrid nodes in the physical space – see Fig. 5B.

Then, the parameterized curves C1ðtÞ and C2ðsÞ are calculated byEqs. (1) and (2) [21], where t and s are the integer variables for thecurve length discretization.

C1ðt ¼ 1Þ ¼ 1C1ðtÞ ¼ C1ðt � 1Þ þ dt; t 2 ½2; . . . ;m�

ð1Þ

C2ðs ¼ 1Þ ¼ 1C2ðsÞ ¼ C2ðs� 1Þ þ ds; s 2 ½2; . . . ;n�

ð2Þ

Fig. 7. The planar advancing displacement profiles in the filling process at 3 ll/min imposed by a syringe pump in the rectangular microchannel with a width of 217 lm.(A) The microchannel geometry and the three studied initial positions of the gas–liquid interface B–D. (B) The displacement of the upstream initial position B of the gas–liquidinterface for Dt = 16.67 ms. (C) The displacement of the downstream initial position C of the gas–liquid interface for Dt = 9.99 ms. (D) The displacement of the downstreaminitial position D of the gas–liquid interface for Dt = 16.67 ms.

Table 2Planar flow rate values and associated errors for the imposed flow rate of 3 ll/min ina rectangular microchannel with 217 lm of width.

Interface location Planar flow rate(mm2/s)

Error (%) Optimized k

B 0.0903 0.013 0.0113C 0.0938 0.010 0.0132D 0.0966 0.070 0.0092


The curvatures kiðCjÞ (i being t for curve C1 and s for curve C2; jassuming the value 1 for curve C1 and 2 for curve C2) of the inter-face locations are calculated making recourse to central differenc-ing through Eq. (3) according to [21].

kiðCjÞ ¼4fyjðiþ 1Þ � 2yjðiÞ þ yjði� 1ÞgDxjðiÞ

fðDxjðiÞÞ2 þ ðDyjðiÞÞ2g3=2

�4fxjðiþ 1Þ � 2xjðiÞ þ xjði� 1ÞgDyjðiÞ

fðDxjðiÞÞ2 þ ðDyjðiÞÞ2g3=2 ð3Þ

In the previous equation DxjðiÞ ¼ 0:5½xjðiþ 1Þ � xjði� 1Þ� and DyjðiÞ ¼0:5½yjðiþ 1Þ � yjði� 1Þ� are the horizontal and vertical half-distancesbetween alternate grid nodes of the same interface location.

It should be noted that there are two kinds of transformationsthat curve C1ðtÞ can experience to become curve C2ðsÞ: a deforma-tion and a combination of translation and rotation.

Once the parameterized domain is defined, the cost function tobe minimized in the search of the optimal path c in the parameter-ized space adopted in the present work, F(s,t), is expressed byEq. (4) [21].

Fðs; tÞ ¼ jktðC1Þ � ksðC2Þj þ k ð4Þ

This cost-function comprises two terms. The first one,jktðC1Þ � ksðC2Þj, is related to the difference of curvatures betweenthe interface locations and weighs the deformation that curve C1(t)experiences to become curve C2ðsÞ .

The second term, k, is a constant that has two roles: a mathe-matical and a physical one.

Knowing that the cost function has always to be positive in theparameterized domain [21], from a mathematical point of view,the k parameter is required to impede the existence of a nil costwhenever any two points of the two discretized curves have thesame curvatures.

From a physical point of view, the parameter k weighs the com-bination of translation and rotation that curve C1ðtÞ can experienceto become curve C2ðsÞ, and its value is determined by an iterativeprocedure as described below in Function Mass_Balance.

3.2.1.2. Function velocity_profile_calculation. This function makesrecourse to other sub-functions represented in Fig. 3B to determinethe interface velocity profile, for a given k.

To determine the optimal path that matches the two curvesBasri and Frenkel [21] proposed the use of the Eikonal equationexpressed by Eq. (5), suggesting the use of the fast marching meth-od [26] to solve such equation as used herein by the sub-functionSolve_Eikonal_Equation.

jrEðs; tÞj ¼ Fðs; tÞ ð5Þ

Fig. 8. Selected images of the flow visualization of the filling process at constantflow rate of 3 ll/min, imposed by a syringe pump, in the rectangular microchannelwith a width of 668 lm, acquired with the CCD camera. (A) View of the emptymicrochannel. (B) Liquid–gas interface position at t = 0 s. (C) Liquid–gas interfaceposition at t = 0.19 s. (D) Liquid–gas interface position at t = 0.41 s.


In the above equation Eðs; tÞ is the cumulative cost in theparameterized domain to go from the origin (point A of Fig. 5C thatmatches the two left extremes of the parameterized curves) to anyother point in the domain, and is defined by Eq. (6) [26]. In thisequation, dr is the discretized arc-length of the a-path for a givencost function F, dr ¼ ðds2 þ dt2Þ1=2, a represents any path definedas a ¼ ½sðrÞ; tðrÞ�, with r 2 ½1;2; . . . ; L� and L is the number of gridpoints that discretizes the a-path length.

Eðs; tÞ ¼Z ðs;tÞ

AFðaðrÞÞdr ð6Þ

The cumulative cost Eðs; tÞ is calculated for all grid nodes of theparameterized domain. The optimal path corresponds to the min-imum cumulative cost value in the parameterized domain requiredto go from point A to B, where B is the point that matches the tworight extremes of the parameterized curves – see Fig. 5C.

Sub-function Optimal_Path calculates the optimal path c in theparameterized domain that has to be orthogonal to the iso-E lines[26] as expressed by Eq. (7) and has as boundary condition Eq. (8).As can be seen from such boundary condition the optimal path isfound starting the procedure at B.

dcðrÞdr¼ �rE ð7Þ

cðr ¼ LÞ ¼ B ð8Þ

The optimal path of the illustrative example is shown in Fig. 5C.

3.2.2. The new functions added to the original Basri and Frenkel [21]algorithm3.2.2.1. Sub-function matching_curves. This sub-function introducesone novelty of this work since to the authors’ knowledge it hasnever been used before. It establishes the correspondence betweenthe two parameterized curves.

In practice, each grid point of the discretized curve C1(t) in theparameterized space is made to correspond univocally to one pointin curve C2(s) based on the optimal path c for each possible cost func-tion – see Fig. 5C. Whenever a point yielded by the correspondenceprocess in the second parameterized curve C2(s) is not a grid nodea linear interpolation between the neighbor grid nodes is performed.

After this, the points of the second interface location are con-verted back to the physical space. Then, the displacement vectorsfor each grid node of curve C1(x1, y1) that is transformed into thecurve C2(x2, y2) in the physical domain are calculated – see Fig. 5D– and divided by the time interval that separates the two interfacepositions. This way, the planar velocity profile is determined.

3.2.2.2. Sub-function mass_balance. This sub-function introducesanother novelty of this work and evaluates the error of the velocityprofile associated with the adopted k value for each cost function F– see Eq. (4) – by performing the planar mass balance expressed bythe continuity Eq. (9) applied to the control surface represented inFig. 5A. This control volume is defined by the side walls, the mostupstream liquid–gas interface location C1(x1, y1) and by anarbitrary section that has to be located downstream the secondliquid–gas interface location C2(x2, y2).

ddT

ZCS

dAþZ

CLðv!� n!Þdl ¼ 0 ð9Þ

In Eq. (9) T is time, v! is the local velocity vector, n! is the local nor-mal to the first position of the interface C1(x1,y1), CL is the controlline and CS is the control surface.

It can be seen from Eq. (9) that the first term on the left-handside does not depend on the determined velocity profile, whereasthe second term exhibits a direct dependence on that parameter.

For each k value a different cost function F, defined by Eq. (4), isobtained, which in turn yields a different optimal path c (Eqs. (6)and (7)) that leads to a different interface velocity profile accordingto the matching procedure explained in the previous sub-function.Therefore, an iterative procedure has to be performed to find outwhich k value yields the minimum error of Eq. (9).

Such error is calculated as the difference between the discret-ized forms of both terms of Eq. (9) expressed by Eqs. (10) and(11), where m and n are the number of nodes used to discretize,respectively, the interface locations C1(x1, y1) and C2(x2, y2).

ddT

ZCS

dA ffi 1DT

Ps¼n�1

s¼1

ðy2ðsÞþy2ðsþ1ÞÞ2 ðx2ðsþ 1Þ � x2ðsÞÞ

h i

�Pt¼m�1

t¼1

ðy1ðtÞþy1ðtþ1ÞÞ2 ðx1ðt þ 1Þ � x1ðtÞÞ

h i

þ y2ðs¼1Þþy1ðt¼1Þ2 ðx2ðs ¼ 1Þ � x1ðt ¼ 1ÞÞ

h i

� y2ðs¼nÞþy1ðt¼mÞ2 ðx2ðs ¼ nÞ � x1ðt ¼ mÞÞ

h i

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

ð10Þ

ZCLðv!�n!Þdl

ffiXm�2

�t¼1

0:5½ðx1ð�t þ 2Þ � x1ð�tÞÞ2 þ ðy1ð�t þ 2Þ � y1ð�tÞÞ2�1=2

~vð�t þ 1Þ� ½�ðy1ð�tþ2Þ�y1ð�tÞÞ;ðx1ð�tþ2Þ�x1ð�tÞÞ�ðy1ð�tþ2Þ�y1ð�tÞÞ2þðx1ð�tþ2Þ�x1ð�tÞÞ2½ �1=2

� �8><>:

9>=>;

�t¼t�1

ð11Þ

The results of this iterative procedure are shown in Fig. 5E andthe value k = 0.0055, corresponding to point P in that figure, is theone that yields the smallest error: 0.05%.

Fig. 9. The planar advancing displacement profiles in the filling process at 3 ll/min imposed by a syringe pump in the rectangular microchannel with a width of 668 lm.(A) The microchannel geometry and the three studied initial positions of the gas–liquid interface B–D. (B) The displacement of the upstream initial position B of the gas–liquidinterface for Dt = 20 ms. (C) The displacement of the downstream initial position C of the gas–liquid interface for Dt = 20 ms. (D) The displacement of the downstream initialposition D of the gas–liquid interface for Dt = 13.33 ms.


Since the input of this algorithm is the output of the previouslydescribed one, all the limitations of the former influence the accu-racy of the results produced by the latter.

Another limitation of the present algorithm, which occurs withany numerical method, is the interface velocity profile dependencyon the grid refinement, i.e. on the number m and n of grid nodesused to discretize the interface locations in the physical domain.This limitation is inherent to the algorithm itself and is alwayspresent even if the results yielded by the image processingprocedure were error-free. However, it should be mentioned thatgrid independence tests performed ensured that the solutionspresented herein for the displacement profiles are all independentof the number of the grid nodes.

Table 3Planar flow rate values and associated errors for the imposed flow rate of 3 ll/min ina rectangular microchannel with 668 lm of width.



B 0.1947 0.024 0.0035C 0.2017 0.047 0.0055D 0.2062 0.044 0.0120

4. Results and discussion

4.1. Application of the techniques to constant rate flows in constant-height microchannels

The algorithm here developed, consisting of the image process-ing technique and the one for the calculation of the velocity pro-files of advancing liquid–gas interfaces described above, was firstapplied to study the filling process of two microchannels with rect-angular cross-sections (with different widths, 217 lm and 668 lmas displayed in Fig. 2A and B) at a constant flow rate of 3 ll/minimposed by a syringe pump.

These flows were chosen for validation purposes since, apartfrom the local effects emerging from surface wall-shape irregular-ities inherent to the manufacture process, the constant flow rate

imposed by the syringe pump is expected to yield a pure transla-tional advance of the liquid–gas interface. Therefore, to assessthe accuracy of the technique, the liquid–gas interface positionsat different moments are determined and their advancing velocityprofiles are calculated. Based on the latter, the planar flow rates areevaluated and compared with each other in terms of both the errorof the mass balance determined by the difference between thevalues yielded by Eqs. (10) and (11) and deviations from the puretranslational movement.

4.1.1. Flow in the constant-height microchannel with a width of 217lm

Fig. 6 shows the results of the image processing technique ap-plied to the filling process of the microchannel with a rectangularcross-section with a width of 217 lm.

The strong hydrophilic character of the microchannel wallsmaterial is clearly discernible from the concave shape of the li-quid–gas interface, which advances faster close to the walls thanin the center (due to the capillarity effect) in spite of the flow being


driven by a syringe pump – see Fig. 6B to D. Experiments revealedthat for the tested flow rate values up to 9 ll/min imposed by thesyringe pump the profile did not become convex.

Fig. 6A shows the microscopic top-view of the empty micro-channel. As it can be seen the microchannel left-wall exhibits asmall shape irregularity. Apart from that fact, that may introducesome local deformations in the velocity profile, it is expected thatthe liquid–gas interface advances mainly with a translationalmovement. Fig. 6B to D allow one to infer that this is the case sincethe interface shape keeps apparently unchanged.

Nonetheless, such translational movement can be confirmedwith the results of the application of the technique to calculatethe interface advancing velocity profiles that are displayed inFig. 7. It should be mentioned that, for the sake of intelligibility

Fig. 10. Sequence of images of the filling process of the triangular microvalve with(B) Time = 0.92 s, 9.5%. (C) Time = 7.05 s, 72.6%. (D) Time = 8.32 s, 85.6%. (E) Time = 8.47 s,(I) Time = 9.717 s, 100.0%.

of Fig. 7B to D, the displacement profiles exhibit only a part ofthe calculated displacement vectors. Otherwise, the figures wouldbecome indiscernible since, for all of them, the spatial resolution,i.e. the largest distance between the origins of two consecutive vec-tors, is only 4.4 lm. Please note also that the interface locationsexhibited are always the most upstream ones used to evaluatethe displacement profiles that are displayed in figures.

It is clear from Fig. 7 that the interface is advancing mostly inthe main flow direction, i.e. with an almost pure translation move-ment, exhibiting small local deformations resulting from theabove-mentioned wall irregularity. These deformations are visiblein Fig. 7B to D: there are small deviations from the verticaldirection of the displacement vectors that indicate either a flowacceleration when pointing to the center as a result of a small

a constant flow rate (3 ll/min) imposed by a syringe pump. (A) Time = 0 s, 0%;87.2%. (F) Time = 8.647 s, 90.0%. (G) Time = 8.977 s, 92.4%. (H) Time = 9.223 s, 94.9%.


section contraction – see Fig. 7C – or a flow deceleration whenpointing to the walls as a result of a small section expansion –see Fig. 7B.

Table 2 contains the planar flow rates calculated for the previ-ously mentioned velocity profiles determined at the interface loca-tions B to D and the corresponding errors evaluated through Eqs.(10) and (11). As it can be seen the planar flow rates are very sim-ilar for the three cases (0.0903–0.0966 mm2/s), which indicatesthat the technique is accurate. In fact, the errors associated to thoseplanar flow rates exhibit similar and very small values for all theprofiles (between 0.01% and 0.07%), which provide a measure ofthe technique accuracy.

Fig. 11. The planar advancing displacement profiles in the filling process of the triangularthe four studied initial positions of the gas–liquid interface B–E. (B) The displacement ofdisplacement of the downstream initial position C of the gas–liquid interface for Dt = 30interface for Dt = 33.33 ms. (E) The displacement of the downstream initial position E o

Recalling that the advancing movements are dominated mainlyby translation – see Fig. 7B to D – the curvatures of the correspond-ing points of the two curves C1(x1, y1) and C2(x2, y2) defining theinterfaces at two consecutive moments are very similar and, there-fore, the optimal k value for the three cases is also similar – seeTable 2.

4.1.2. Flow in the constant-height microchannel with a width of 668lm

Fig. 8 shows results similar to those of Fig. 6, but for a widermicrochannel with a 668 lm width. Again, the hydrophilic charac-ter of the microchannels material is discernible from the concave

microvalve under an imposed flow rate of 3 ll/m. (A) The microvalve geometry andthe upstream initial position B of the gas–liquid interface for Dt = 16.67 ms. (C) The.0 ms. (D) The displacement of the downstream initial position D of the gas–liquid

f the gas–liquid interface for Dt = 30 ms.

Table 4Planar flow rate values and associated errors for the imposed flow rate of 3 ll/min forthe variable height rectangular microvalve (with vertical expansion).



B 0.2 0.05 5.5 � 10�4

C 0.4 1.06 5D 0.97 41.6 100E 0.22 0.05 2.1 � 10�3


shape of the liquid–gas interface, although not so pronounced as inthe previous case due to the much larger width of themicrochannel.

As in the previous case, the microscopic top-view of the emptymicrochannel displayed in Fig. 8A also reveals that its left-wallexhibits a small shape irregularity and, therefore, it is expectedthat the liquid–gas interface advances mainly with a translationalmovement, which is inferable from Fig. 8B to D since the interfaceshape remains apparently unchanged.

Such translational movement is confirmed in Fig. 9 that depictsthe planar advancing displacement profiles and reveals an inter-face advance mostly in the main flow direction. Like the case be-fore, there are small local deviations of the displacement profilesresulting from the wall irregularity, yet less pronounced in thiscase than in the previous one: vectors deviating from the verticaldirection and pointing to the microchannel walls result from aquite small expansion and indicate a very small flow deceleration.As before, the displacement profiles exhibit only a part of the cal-culated displacement vectors to avoid the figures to becomeindiscernible.

Table 3 shows that the values of the planar flow rates calculatedfor the velocity profiles at the interface locations B to D are verysimilar for the three cases (0.1947–0.2062 mm2/s) confirming theaccuracy of the technique, as do the corresponding errors thatare similar and very small for all the profiles (between 0.024%and 0.047%).

The optimal k value follows the same pattern as in the previouscase.

4.2. Limitations of the algorithm: application of the techniques to aconstant rate flow in a triangular microvalve with variable height

Figs. 10 and 11 exhibit the results of the algorithm applied tothe flow filling process in the triangular microvalve of Fig. 2D witha vertical expansion (in the zz direction of Fig. 2) at constant flowrate imposed by the syringe pump (3 ll/min). This valve cavity(height larger than that of the upstream microchannel) is notice-able in Fig. 10 through the gray-color triangle.

Fig. 10 exhibits several locations of the liquid–gas interface inthe filling process of the microvalve under study obtained withthe image processing technique algorithm. The experiments evi-dence marked flow asymmetries. At the expansion section –images (C and D) – the interfaces appear somewhat symmetrical,which is due to the quite slow advance of the liquid–gas interfaceat that location imposed by the sudden expansion. This flow decel-eration is due to the surface tension effects in the yy direction thatact in opposition to the flow advancement as a consequence of theinversion of the interface concavity (from concave to convex),which occurs to allow the fluid to enter the microvalve.

After that – images (E to H) of Fig. 10 – the asymmetries becomeobvious. This phenomenon is most certainly related to the effect ofthe wall irregularities (that alter locally the interface contactangle).

At the sudden expansion of the channel cross-section there is adecrease of the interface contact angle (that is the basis of opera-tion of capillarity-driven stop microvalves [7]) due to the above-mentioned inversion of the interface concavity. Even though, dueto surface irregularities there may be a flow advance obstructionin a determined location whenever the critical contact angle keepsits value above the interface contact angle. This appears to havebeen the case occurred at the microvalve right-hand side in image(E) that makes the flow asymmetrical.

Fig. 11 displays the velocity profiles of the advancing interface.Image (A) of this figure displays the upstream liquid–gas interfacesat four different positions, B, C, D and E, for which the displacementprofiles are determined. As for the previous cases and for the sake

of intelligibility of Fig. 11B to E the displacement profiles exhibitonly a part of the calculated displacement vectors.

Table 4 shows the planar flow rate values and the correspond-ing errors of the interface advancements from locations B to E. Dif-ferently from that occurred for the previous cases, in the flowzones associated to the vertical expansion of the valve (interfacelocations C and D), where a marked vertical velocity componentis present, the mass flow rate values are considerably different(0.4 and 0.97 mm2/s) from that at interface B (0.2 mm2/s). Relatedto this, the errors associated with the planar flow at interface loca-tions C and D exhibit much larger values (1.06% and 41.6%) thanthose obtained for the regions where the velocity profile has nosignificant vertical component (interface locations B and E witherrors of 0.05%).

This clearly shows that the technique proposed herein is hardlyapplicable to flows where the vertical velocity component is rele-vant since it is unable to capture the interface movement in thatdirection.

Finally, the method accuracy can be once more confirmed bycomparing the mass flow rate values and the corresponding errorsin the sections where the vertical velocity component is not rele-vant – interfaces (B) and (E). As revealed by those values in Table4, the mass flow rates are quite similar (0.2 and 0.22 mm2/s) andthe error is quite small (0.05%).

5. Conclusions

An experimental method to measure the velocity profiles of themoving interface of unsteady two-phase (liquid–gas) flows insidemicrodevices was presented. The method uses microflow visuali-zation performed with a high-frequency CCD camera connectedto a microscope to acquire flow images. Processing digitally theseimages with a defined algorithm one can obtain the mathematicalfunctions defining the liquid–gas interface locations at each frameof the acquired images with an accuracy of ±½ pixel. Another algo-rithm based on the curve matching technique was used to evaluatethe velocity profiles of the advancing interface. This algorithmincludes two new functions: the first one defines the velocityprofile from the optimal path obtained in the matching procedureof two parameterized positions of the moving interface (and isapplicable to open curves typical of advancing liquid–gas inter-faces in confined microflows); the other function performs anunsteady mass balance viewing the determination of the best costfunction for the matching procedure that yields the velocity profilewith the least mass imbalance.

Application of the measuring technique to flow situations ofconstant rate imposed by a syringe pump in simple geometries(microchannels of constant height without vertical velocity com-ponent), yielded accurate results, showing an expected almostpure translational movement at constant flow rate with a massimbalance below 0.07%.

On the other hand, the results of the technique applied to a tri-angular microvalve with vertical expansion (the microvalve heightbeing larger than that of the preceding microchannel) showed thatthe technique is hardly applicable to flows where a verticalvelocity component is relevant.


Acknowledgements

The authors are grateful the financial support provided byFundacao para a Ciencia e Tecnologia for the research projectPTDC/EME-MFE/099696/2008. The authors also acknowledge theproject partner Biosurfit for the provision of the chips with mic-rodevices for the visualisation tests, and to Prof. Antonio Moreiraof IN+ for the permission to use the Phantom V4.2 CCD camera.

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