Microdrops in Microfluidic: Estimation of Geometrical Form
Chekifi. T1, Dennai. B, Khelfaoui. R
Laboratory ENERGARID, University Tahri Mohammed Bechar, mechanical engineering department. B. P.
417, route de Kenadsa, 08000 Bechar, Algeria chekifi.tawfiq@gmail.com2
ABSTRACT
Manipulation of small-sized liquid quantities is one of the key issues in technology, biology, chemistry and medicine.
Where the increased surface-to-volume ratio making surface effects more dominant [42], which allows to reduce time of reaction and economies the costs of manufacturing products. The microdrop is a typical example of a small-sized liquid quantity; that can be considered as samples, reagents and supplementary indicator to approach us in the direction of reaction and the state of microdrop. The creation of reproducible in stable geometrical form of microdrop makes a critical step during the manufacturing of microdrop. Pressure based manufacturing of microdrop nowadays a common way to create reproducible and stable small-sized drops. Many geometrical form of microdrop can be generated in different techniques. Proofed that microdevices based on pressure based manufacturing of microdrops has already been successfully implemented. This paper serves as a review of the most important contributions, in the development of the creation of reproducible and stable microdrops in the past decades. Besides serving as a reverence for researchers in this area, this paper is also a resource for everybody who wants to identify the easiest way to precise microdrops geometrical form for a particular application by simple analysis.
KEYS WORDS
form, microdrop, description and theoretical analysis.
1*Corresponding author
ENERGARID Laboratory, University Tahri Mohammed Bechar, mechanical engineering departement B. P. 417, 08000 Bechar, Algeria Tel.: 213 49 81 55 81/91, fax: 213 49 81 52 44
E-mail address: chekifi.tawfiq@gmail.com
NOMENCLATURE L: length of canal (m)
dh: hydraulic diameter of canal (m)
A1: constant of equation (non-dimensional number) specify for each microsystem.
A: Surface
Ca : capillary number (non-dimensional number) d: diameter of canal (m)
f: frequency (1/s) h: depth of channel(m) L: drop length (m) Q: flow rate (μL/min)
Re : Reynolds number (non-dimensional number) V: Volume (m3)
W1: width of channel (m) W: Work energy (j)
σ: interfacial tension between two phase (N / m) α2: parameter depend of flow rate of two phases α, β: coefficient adjusted by experimental values ɛ, ɷ, k: parameters depend of geometric system μ: dynamic viscosity (Pa.s)
ρ: density
U : velocity of dispersed phase (m / s)
1. INTRODUCTION
Due to the great interest of researchers in several field and the competition of enterprises and manufactories, rapid development of microfabrication technologies has facilitated a broad range of microfluidic applications, especially in life sciences [40], as chemistry, biology and medicine and divers rang application. The manipulation of small volumes is also simplified, Indeed, drops provide new physical and chemical contrasts with the outer medium; such as, the dielectric constant or the interfacial tension, which can be used to manipulate the minute volumes on-chip [39] The minimum volume for a given liquid in microfluidics is a microdrop, which can be obtained by several techniques.
Many microfluidic devices have been designed to generate uniform microdrop, including geometry dominated devices [38,37], flow-focusing devices [36–35], T-junctions [34–33] and co-flowing devices [32,31]. Biological reactions tend to be especially difficult to control, but microfluidic systems offer means to improve the control and the efficiency for even the most complicated biological assays [30]. Microdrops show great promise as a new high-throughput technology in chemistry, biochemistry, and molecular biology.[ 29-28].
On the other hand, fluid dynamical in this field is very sensible, where miniature moving or vibration can modified the development of operation which can be lead to false results, so it is a challenge to have correct local measurement by experimental devices, that’s why we need any meaning of measured parameters without contact with the microsystem. The behavior of microdrop geometrical form may give us several ideas and meanings about its stability, its equilibrium which means existing of pressure or force surround it, its concentration in existing of reagents and the interaction between molecules; for example, study of microdrops evaporation has a direct relation to variation of its geometrical form.
A large body of work has recently attempted to tackle these fluid dynamical questions, leading along the way to creative new designs for microfluidic systems and new physical approaches to control the behavior of drops[1], the microdrops are also often studied by numerous researchers, some of them have taken it by experimental methods, others have treated it by analytical methods and others have married the two methods to give more reliability. After broad examination of these works, this work limited some models which can help us to find a mechanism of variation of microdrop shape, where the latter depends on the state of microdrop, that exists in two cases, the first is when the microdrop is in static state characterized by contact angle; whereas the second is when it is in dynamic state, which is function of different parameters, such as viscosity of fluid, diameter of channel, non dimensional numbers and others. basing on the logical analysis of the equation, which dominate on the state of each case, we found several probability of geometrical form.
This subject has been already treated by other angles and different methods. In this paper we focus our interests on logical reasoning, in order to analyze the variation of geometrical form of microdrop in function of different parameters, by the use of some of the previous models, which are involved in the running of this phenomenon.
2. MATHEMATICAL FORMULATION OF MICRODROP DIAMETER
Since the first creation of microdrop by Thorsen (2001), microdrops formation has been widely studied in various microgeometries, such as microchannel arrays, [13] [14] T-sections, [15] [16] hydrodynamic focusing, [17] membrane emulsification, [18] flow focusing, [19] [20] and concentric injection [21] [22].
The authors have tried to describe and equate this phenomenon, because it has a direct relation to diameter of microdrop, among them have based on their experimental results, others have developed older models (Tab1) and have extended to more parameters. Each model is applicable under limited condition, as the geometry of production and its dimensions, the fluid used to represent the two phases and their flow rates, the capillary number and other non-dimensional numbers. The models which we found in this research are summarized in the following table.
Table 1: The equation of microdrop production procedure in chronological order
In this table the models are established by empirical and experimental basics, one of them has developed, improved and generalized old model to a new one that includes more parameters, which coming at this operation. Each author has obtained its equation in limited condition as value of capillary number, dimension of channels, regime of flow, viscosity of fluid and type of microsystem.
In our work we have chosen the last three younger models to study the microdrop shape through, it diameter, it length and it volume. The use of these models allows us to compare obtained results, weher each one of them includes parameters that are different from the others.
3. GEOMETRICAL FORM OF MICRODROP
According to different objects of the microdrop exploitation. there are two important cases when the form of microdrop is necessary in different application objects. Firstly, when the microdrop is in static state; for example, it can be used in lab on chip application where it contains sample of reaction. Secondly, when it is in flowing as the case of manufacturing. To get precised and exact information about the action into the microdrop, especially during the manipulation is too difficult. For this reason, the Change of microdrop shape may be occured in short time, which signify an action around or into microdrop. The use of the other microdevices can upset the result during the measurement, this is why we need any indicator to know at least if the microdrop is in equilibrium, or it submits to a pressure or force in the both states: the dynamic state and the static one.
3.1 Microdrop Form at Static State
For several operation the microdrop must be in static case, as reaction of chemistry. When the microdrop is in equilibrium state, the parameter which allows to describe the microdrop form is contact angle, which
Author Equation Interval of verification
Thorsen (2001) [2] dg/dh = 1/Cac “dripping regime “, T junction
geometry
W/O, 0.01 < Ca < 0.3 Garstecki (2006) [3]
Lg/w1 = 1 + α(Qd/Qc)
“slugs regime“, T junction geometry
Ca < 10-2 , µc>µd Tan (2008) [4]
Lg/w1 = k(Qd/Qc)α(1/Ca)β
“slugs regime“ in cross junction geometry for the both emulsion O/W and W/O
Ca < 10-2 , µc>µd
Xu (2008) [5]
Lg/w1 = ε + ω(Qd/Qc)
“slugs regime“ in T junction geometry
Ca < 0.002 , µc>µd Lg/w1 = ε + k(Qd/Qc)α(1/Cac)β
“slugs regime” in cross junction geometry
0.002 < Ca < 0.01, µc>µd dg/dh = 1/Cac. (w1h – π/4.dg2)/w1h
“dripping regime “ ,T junction geometry.
W/O
0.01< Ca < 0.2
Cubaud (2008) [6] μd/μc > 20
Ca<10-2 DI MECELLI (2008)
[10]
Cag varies between3.8 x 10-4 and 2.8 x 10-2
Alain (2009) [7] L/dh < 3, chip in glass-silicum,
capillary reactor, PDMS
depends on the wettability of a surface which is also depend on the cohesion of forces and the adhesive of forces (Fig.1).
Figure 1: Illustration for the derivation of the static contact angle θ . The droplet wets the area A on a surface.
An energy minimum is reached when dW /dA = 0. The surface area remains stable at this minimum [8].
A droplet sitting on a substrate has three different phase boundaries: the solid-liquid interface with the interfacial energy σSL, the solid-vapor interface with the interfacial energy σSV and the liquid-vapor interface with the interfacial energy σLV. The next Figure illustrates the derivation of the contact angle θ [8]. The virtual displacement of the contact line and its resulting variation of the free energy W is [9]:
dW = σSL dA - σSV dA + σLV cos θ dA (1) The minimum energy is reached when dW /dA = 0. Solving this equation for cosθ leads to the Young equation:
(a) Wetting surface – contact angle is (b) Non-wetting surface – contact θ <90° angle is 90° < θ <180°
Figure 2: Two examples of the wetting behavior of liquids (wetting & non-wetting surfaces) [7].
The contact angle θ at a solid-liquid interface is a measure of the degree of wettability. If the angle lies between 0° and 90°, the surface is partially wetted by the fluid (see Fig.2a). If it is between 90° and 180° it is called non-wetting (Fig.2b) [8].
According to Young equation Eq.(2), the geometrical form of microdrop at static state depends on the contact angle. For this case the microdrop will be tended into three main geometrical forms:
- Spherical form for 90° < θ <180°
- Ellipsoidal form for 45° < θ <90°
- Plated form for θ <45°
It is clear that to get spherical form, the contact angle must be increased (90° < θ <180°), where the interfacial energy (σLV) must be also increased; furthermore, the difference between the interfacial energy solid-liquid (σSL) and the interfacial energy solid-vapor (σSV) should be negative. These conditions imply to use a rough substrate.
3.2 Microdrop Form in Dynamic Flow State
After the production of the droplets, they are transported along microfluidic channels by the carrier fluid [1], refering to the model which we have chosen to study in this case, the geometrical form of microdrop during the manipulation of microdrop in flowing depends on the drop volume, the channels diameters, the frequency of the produced drop, the flow rate, the capillary number of the continuous phase and the Reynold number of the dispersed phase, each one of these parameters has an influence on the drop shape. The analysis of geometrical shape of microdrop is essential for many applications, where we need to get some information during the flowing; such as, measuring of velocity, mixing during droplet merging and fusion of two microdrops.
3.2.1 Analysis of DI MECELI RAIMONDIE 2008 Model[10]
Despite the fact that DI MECELI RAIMONDIE [10] provides an equation that seems simple, it shows best agreement between the calculated results and the experimental results. This model links the production frequency (fg) with the microdrop volume (vg) to have the flow rate (qd). The microdrop volume may give us
(2)
an idea about the geometrical form. In the next analysis we study the effects of the flow rate of the dispersed phase and the drop frequency on the drop volume in T-junction geometry. As usual, we use the T-junction geometry where there are: the channel of dispersed phase carries the water and the channel of continuous phase carries the oil.
[10] (3)
Figure 3 : (color online) The variation of the microdrop Figure 4 : (color online) The variation of the frequency (fg) volume (vg) in function of flow rate (Qd) for in function of flow rate (Qd) for four volumes.
three frequencies (fg).
As shown in (Fig.3), the microdrop volume is dependent on the flow rate, which is varied over a broad range (from 0 to 0.05 micL/min). We fixed three frequencies (f1=2000, f2=1000 and f2=100) to remark their influence. For the high frequencies (f1=2000 and f2=1000), we observe that the drop volume increased by the increase of the flow rate in a linear relation. For the low frequency, we have an important development of the drop volume, so the frequency affects on the microdrop volume in reverse order .
To estimate the geometrical form of microdrop through this model, we have selected four cases to analyze it:
- The first case: Low frequency with low flow rate: Means that few microdrops which flow with small volume, signify that the volume of continuous phase occupies the most of channel volume (vc>>vg), each one is surrounded by an important volume of continuous phase, which apply high pressure toward the centre of microdrop and leads to a stable flow of microdrop. For this case, the friction between microdrop and wall is neglected and leads to spherical form.
- The second case: High frequency with low flow rate: The microdrop has the smallest volume
; consequently, the continuous phase flow in full of microdrop, the probability of collusion is very high, it can be led to many fusion. In this case, the form geometrical is instable.
- The third case: High frequency with high flow rate: The volume of microdrop is bigger than the previous case, the possibility of fusion is the highest in existing of the the channel friction, which leads to elongated form.
- The fourth case: Low frequency with high flow rate: The volume of microdrop is the biggest compared with the previous case, the microdrop flow in contact of wall channel, in which the friction will be created, which leads to a deformable various shape, in particular, we can have elongated form for the lightest frequency and the highest flow rate, where the volume drop is too important.
The (Fig.4) shows the influence of flow rate of dispersed phase (Qd) (from 0 to 0.05 micL/min) and the volume of microdrop ( vg1=1x10-4, vg2=2x10-4,vg3=3x10-4,vg4=4x10-4) on the frequency of production. It can be clearly seen that the effects of (Qd) and (vg) are inversed. In order to get low frequency(fg) we should have weak flow rate and produce small volume, while to attain high frequency, we should increase the flow rate and decrease the volume of microdrop.
The model of DI MECELI RAIMONDIE (2008) [10] offers a possibility to estimate and dominate the geometrical form by simple and logical reasoning, where the flow rate (qd) is in function of microdrop volume (vg) and production frequency (fg). This model also allows to govern the frequency, which affects directly on the volume and the geometrical form of microdrop.
3.2.2 Analysis of MARCATI 2009 Model [7]
After the examination of the equation which allows to describe the phenomenon of microdrop generation, we have chosen the model of MARCATI (2009) [7] Eq.(4), which is considered one of the most important and
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0 1 2
x 10-4
flow rate (micL / min)
volumedrop(micL)
f1=2000 f2=1000 f3=100
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0 1000 2000 3000 4000 5000
flow rate (micL / min)
frequency (Hz)
Vg1=0.00004 Vg2=0.00003 Vg3=0.00002 Vg4=0.00001
1 1.5 2 2.5 3 3.5 4 4.5 5 x 10-3 1
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Cac
L/dh
µc2, dc/dd=8 µc1, dc/dd=8 µc2, dc/dd=4 µc1, dc/dd=4 µc2, dc/dd=1.2 µc1, dc/dd=1.2
1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3 1.4
1.6 1.8 2 2.2 2.4 2.6
Cac
L/dh
µc2, Red=90 µc1, reynd=90 µc2, Red=5 µc1, reynd=5
efficient models. Its validity is large than the old one and its results in the experimental and the theoretical are very acceptable. Besides, it covers more parameters which participate in the operation, it is applicable for several microsystems, as T-junction geometry in silicon glass for (water / silicon oil) system, cross junction geometry in silicon glass for (water / silicon oil) system, capillary cross junction in round section geometry for (water / silicon oil system), and others.
During the analysis of this model, by taking in consideration all the conditions where the model is valid. We study the T-junction geometry, where µc1 (viscosity dynamic of silicone oil is superior than µc 2 (viscosity dynamic of gas oil), which help us to find a description of microdrop variation form in function of the main parameters, that affect on the geometrical form of drop, by the use of polynomial regression of the fourth degree order.
For this model, some effects can be strong or weak depending on the details of the flow, which are commonly defined by the geometrical and physical parameters including the capillary number of continuous phase (Cac), the viscosities of the both phases (µc, µd), the Reynold number (Red) of dispersed phase and the diameter channel (dc, dd).
In particular, the capillary number describes the relative importance of viscosity and surface tension. It is defined as Ca = μU/γ where μ is the dynamic viscosity, U is a typical velocity of dispersed phase and γ is the surface tension coefficient. In the flows where one fluid replaces another, the coherence or breakup of the interface will depend strongly on Ca, hence on the velocity of the moving interface. See [26] for a good review of the motion of drops in capillaries and the works of groups [23, 24, 25], which study different geometries in different values.
[7] (4) As shown the next (fig.5.6), we have chosen some values to check the influence of each parameter on the ratio of microdrop length to hydraulic diameter, three values of diameter ratio (dc / dd=8, 4 and 1.2) are controlled, two different Reynold numbers (Red= 90 and 5) are fixed and two different viscosities for the continuous phase (µc1=34 µc2=12) are examined.
Figure 5: (color online) The variation of the ratio of drop Figure 6: (color online) The variation of the ratio of drop length to hydraulic diameter in function of length to hydraulic diameter in function of capillary number for three diameters of capillary number for three Reynold numbers.
T-junction ratio.
We can easily observe in (fig.5.6) that in order to decrease the microdrop length we have many possibilities:
The first is to decrease the capillary number of continuous phase, which means to increase the shear stress (Ca = μU/σ). The second is to decrease the Reynold number of dispersed phase (inertia forces, Re = ρUdh/μ).
The third is to use two diameters of microsystem near the unit.
The (fig.7) plots the non-dimensional microdrop length of as a function of Reynold number of dispersed phase at Ca= 0.001 and 0.003.
In this model the essential principle, which help us to estimate the geometrical form of microdrop is:
When the ratio of microdrop length to hydraulic diameter of channel is near to the unit (L / dh 1), in which it can be deduced that the geometrical form tends to the spherical form, otherwise (L / dh >>1) in which the geometrical shape will tend to elongated form.
- The capillary number (Cac) allows to compare the shear stress of continuous phase to the interfacial tension of dispersed phase. In this equation the (Cac) characterize the effect of the continuous phase. When it is low (Cac= 0.001) (Fig.5.6) means that the shear stress applied by the continuous phase on the interface of dispersed phase will be weak, in which to deform the volume then generate the microdrop it needs a long instant: therefore, an important volume of the dispersed phase will penetrate the channel (L / dh >>1); at this moment, the flow of carried liquid make the volume in elongated form before its generation. We note that in this case it is so possible to have a thread.
In the opposite case, when the (Cac) is high (Cac=0.004), it indicates that the interfacial tension is decreased, which permits to the shear stress of continuous phase to deform the dispersed phase easily in short time, which allows to a small volume (L / dh 1) to be generated taken spherical shape.
- To dominate the capillary number (fig.5), we have a possibility to change diameter ratio, where high diameter ratio (dc / dd=8) means high diameter of continuous phase, which implies low flow velocity, that leads to low Cac creating an elongated form.
- Flow of fluids in microfluidic systems are usually characterized by low values of the Reynold number (Re
= ρ ul/m, with ρ and m being the density and the dynamic viscosity of the fluid respectively, u the speed of flow, and l the characteristic dimension of the system). For Re << 1 flow is dominated by viscous stresses and pressure gradients - inertial effects are neglected - and the trajectories of fluidic particles can be controlled precisely [27]. In this model, Reynold number characterizes the dispersed phase channel.
High value of Reynold number (Fig.6.7), means that the inertia forces is increased, which characterize the velocity of dispersed phase to enter the channel of the continuous phase until the generation of microdrop, when this velocity is important, the shear stress applied by the continuous phase will deform a big volume to generate drop, which allows to get an important drop length, where the ratio of drop length to hydraulic diameter will be important (L / dh >>1), which means that the microdrop will have a longer shape, on the other hand, the drop length will have a small drop length (L / dh 1), with spherical form.
- The influence of the viscosity of continuous phase affirms the influence of the capillary number. This influence appears in all the plotted figure in basic of MARCATI model [7], when the viscosity is increased means that the shear stress which is applied by the continuous phase tends be important than the interfacial tension, which allows to deform the dispersed volume easily leading to small microdrop (L / dh 1) in spherical form, compared with slow displacement which is the opposite case that produce the elongated form.
3.2.3 Analysis of Cubaud and Mason 2008 Model [6]
We have selected the next model Eq.(5.6), which describe the production of microdrop in cross junction, compared with the previous models, this model has applied in square channel for two case (d / h) inferior to 2.5.h and d / h higher than 2.5.h, The results of this model are obtained by using an universal hydrodynamic focusing geometry [41], into a square outlet channel with straight walls. This geometry is characterized by only one length scale h and significantly differs from a flow-focusing geometry, which includes a constriction at the entrance of the outlet channel. Here, in the absence of defocusing effects downstream from a constriction, a study of the influence of fluid and flow properties on droplet and slender viscous structure formation in a square microchannel is presented [6], in this model the geometrical shape depends strongly of
0 100 200 300 400 500 600 700 800 900 1000
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Red
L/dh
µc2, ca=0.001 µc1, ca=0.001 µc2, ca=0.003 µc1, ca=0.003
Figure 7: The variation of the ratio of drop length to hydraulic diameter in function of the Reynold number for the three capillary numbers.
1 1.5 2 2.5 3 3.5 4 4.5 5 x 10-3 -1
0 1 2 3 4 5 6 7
Cac
d/h
q1=1 q2=2 q1=2 q2=2 q1=2 q2=1
1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3 1.4
1.5 1.6 1.7 1.8 1.9
2x 10-3
Cac
d/h
q1=1 q2=2 q1=2 q2=2 q1=2 q2=1
capillary number of second liquid in microsystem and the factor α2 which is function of flow rate of two liquid, as shown in the next equation (Eq.5.6).
Using the same principle to get ideas about geometrical form of microdrop, we study the variation of ratio of microdrop diameter to width channel (d / h) as function of capillary number of continuous phase and the flow rate ratio. This model differ two case, d > 2.5h and d < 2.5h.
Figure 8: The variation of ratio of diameter drop to depth Figure 9: Thevariation of ratio of diameter drop to
channel in function of capillary number for three depth channel in function of capillary number low rate of both channel (d > 2.5h) for three flow rate of both channel (d < 2.5h)
As presents the (fig.8), we have examined the influence of the capillary number for three cases (q1 > q2, q1=q2 q1 < q2). In this model, that the author left some doubt about the validity of this model, as we found the negative ratio of the drop length to depth channel (d / h < 0), this equation illustrates a good agreement with MARCATI (2009) model [7].
- For (d > 2.5h), the model presents double effects of the capillary number on the geometrical form, in which a low capillary number means that the shear stress which is applied by the continuous phase will be low.
That’s why we observe that the value of (d / h) is big compared with the obtained results in MARCATI (2009) model [7], where the shear stress needs a long term to create a microdrop; consequently, the microdrop tends to take the elongated form; for instance, for a low capillary number (0.001), the ratio (d / h) attains high values (3, 4, 5, 6), which indicates that the microdrop have a length so bigger than its width means the elongated form; even as, when (d / h) approachs to the unit it tends to take spherical form. For this case, the important values of the flow rate of the both continuous phases affect directly on the stability of the shape; Furthermore, to obtain the smallest diameter microdrop in the elongated form, all we have to use (q1 < q2), in this case, the flow rate of the both continuous phases can be an important advantage to make the microdrop on stable flow in the channel axis, this flow allows to have a thin layer on the wall, which prevents the contact with the wall and keeps the shape in equilibrium state.
- For (d < 2.5h) as plots the (fig.9), the ratio is too small (d / h <<1 in the order of 10-3), means that the diameter microdrop is too small compared with the width channel, which leads to make the microdrop surrounded by an important continuous phase volume; therefore, the microdrop takes spherical shape in all the cases, at this point, the flow rate doesn’t have influence on the shape, we have to make separated microdrops and avoids the fusion in aid of DI MECELI RAIMONDIE (2008) model [10], which allows to dominate the production frequency, the low frequency leads to stable geometrical form. It can be clearly observed that the effect of capillary number in cross junction is the same for T-junction.
[7]
4. CONCLUSION
The development of micro- and nanotechnologies has facilitated the precise manufacturing of microfluidic devices.[11],[12], this work represents a vision and a platform of the mechanism of variation of the microdrop form for static and dynamic state. In this domain of research especially in experimental method, we need any meaning to get and affirm measurement, where the manipulation of microdrop is sensible to microvibration and micromoving, which can easily change and upset the behavior of microdrop. Researchers in this field have developed a variety of different techniques to control the action into microdrop behavior. The identification of the shape of microdrop is too necessary;
in which, some operations are extremely adequate in spherical form, other are particularly appropriates in elongated shape as splitting; furthermore, the variation of microdrop shape can be an efficient indicator.
For the static case, the geometrical form of microdrop can be estimated in the basic of contact angle Eq.(2), the latter is function of the different interfacial energies (solid-vapor, liquid-vapor and solid-liquid), the main parameter affecting desired microdrop shape is interfacial energy for (solid-liquid), which allows to change and modify the type of substrate (rough or smooth), three main shapes are possible to make the microdrop in platter form we have to get (θ <45°), which means to reduce the interfacial energy (liquid-vapor); as result, use the smooth substrate, whereas the increase of contact angle leads to ellipsoidal, then spherical form.
For the dynamic state, we have found several possibility to estimate and dominate the geometrical form. The main parameter which aid us to descript the variation of microdrop shape is that evaluates the microdrop dimension to channel dimension Eq.(4.5.6), the microdrop tends to have spherical form when it size tends to decrease compared with channel size.
However, we have found that the capillary number, which characterizes the continuous phase plays an important role to identify the microdrop shape, in which it allows to dominate the microdrop diameter; for example, to get spherical form (means high ratio L / dh) we should decrease Cac. In addition, the Reynold number of dispersed phase can play the same role of capillary number; moreover, we have seen others parameters which influent at the form, as well as, liquid viscosity, for cross junction geometry, flow rate of two lateral channel plays an important role, on the stability and equilibrium of elongated microdrop, it allows to negligee the effect of the friction channel on microdrop.
In aid of an important equation Eq.(3) which allows to find volume and frequency microdrop, the increase of volume for high flow rate make the microdrop in instable state, as well as the high frequency leads to possibility of fusion of several microdrops, means to dominate indirectly the microdrop shape.
In this paper we have tried to provid this young and important field of research with more information about the geometrical form of microdrop, which still hasn’t take place in the microdrop field yet,
We note that in the survey presented in this paper, we supposed that the microdrop is homogeny, It is an interesting question if—and how— different molecular, particular concentration into the microdrop will modify its shape, especially for dynamic case.
Finally, better understanding of this important subject, needs to be checked by numerical, experimental method to be more reliable and exploitable for different uses.
INDEX
1: liquid of the first channel 2: liquid of the second channel c: continuous phase
d: dispersed phase h: hydraulic g: drop L: liquid S: solid
V: vapor calc : reference to calculate size.
SHORT CUTS
DEP: dielectrophoresis MHD: magnetohydrodynamic PDMS: polydimethylsiloxane UV: ultra-violet
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