Study for optical manipulation of a surfactant-covered droplet using lattice Boltzmann method Se Bin Choi, Sasidhar Kondaraju, and Joon Sang Lee Citation: Biomicrofluidics 8, 024104 (2014); doi: 10.1063/1.4868368 View online: http://dx.doi.org/10.1063/1.4868368 View Table of Contents: http://scitation.aip.org/content/aip/journal/bmf/8/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Binary droplet collision simulations by a multiphase cascaded lattice Boltzmann method Phys. Fluids 26, 023303 (2014); 10.1063/1.4866146 Simulations of Janus droplets at equilibrium and in shear Phys. Fluids 26, 012104 (2014); 10.1063/1.4861717 Droplet size distributions in turbulent emulsions: Breakup criteria and surfactant effects from direct numerical simulations J. Chem. Phys. 139, 174901 (2013); 10.1063/1.4827025 Viscous coalescence of droplets: A lattice Boltzmann study Phys. Fluids 25, 052101 (2013); 10.1063/1.4803178 Droplet spreading on a porous surface: A lattice Boltzmann study Phys. Fluids 24, 042101 (2012); 10.1063/1.3701996
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BIOMICROFLUIDICS 8, 024104 (2014)
Study for optical manipulation of a surfactant-covered droplet using lattice Boltzmann method Se Bin Choi,1 Sasidhar Kondaraju,2 and Joon Sang Lee1,a) 1
Department of Mechanical Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea 2 Department of Mechanical Engineering, Indian Institute of Technology, Block III #174, Main Building, New Delhi 110016, India (Received 10 January 2014; accepted 3 March 2014; published online 12 March 2014)
In this study, we simulated deformation and surfactant distribution on the interface of a surfactant-covered droplet using optical tweezers as an external source. Two optical forces attracted a single droplet from the center to both sides. This resulted in an elliptical shape deformation. The droplet deformation was characterized as the change of the magnitudes of surface tension and optical force. In this process, a non-linear relationship among deformation, surface tension, and optical forces was observed. The change in the local surfactant concentration resulting from the application of optical forces was also analyzed and compared with the concentration of surfactants subjected to an extensional flow. Under the optical force influence, the surfactant molecules were concentrated at the droplet equator, which is totally opposite to the surfactants behavior under extensional flow, where the molecules were concentrated at the poles. Lastly, the quasi-equilibrium surfactant distribution was obtained by combining the effects of the optical forces with the extensional flow. All simulations were executed by the lattice Boltzmann method which is a C 2014 AIP Publishing LLC. powerful tool for solving micro-scale problems. V [http://dx.doi.org/10.1063/1.4868368]
I. INTRODUCTION
Understanding the rheological behavior of emulsion systems has a broad range of industrial applications, in the food, medical, cosmetic, polymer, water purification, and pharmaceutical industries. An emulsion is defined as a mixture of two or more liquids that are immiscible, such as water-in-oil (W/O) or oil-in-water (O/W). A number of products are made up of droplet-based immiscible mixtures (emulsion). Surfactants which adhere to the droplet interface and reduce its surface tension are mainly used to obtain the stability of emulsion and used to control both the droplet size and deformability. Polymer manufacturing industries strive to provide enhanced methods of emulsion control in order to satisfy market demand. The behavior of surfactant-covered droplets in simple flow has been studied experimentally and numerically.1–11 The deformation and burst of small droplets in steady 2-dimensional linear flows were experimentally investigated, comparing the results with predictions of several available asymptotic deformation and burst theories, as well as numerical calculations.1 The effects of surfactants on drop deformation and breakup in extensional flows at low Reynolds number were discussed. The analytical and numerical results indicated that surfactants behave differently at low and high capillary numbers, which leads to different levels of deformation.2 The drop breakup mechanism in extensional flow caused by the transport of surfactant molecules on the interface was numerically simulated, which results in changes in local surface tension.3 The influence of deformation and de-aggregation of droplets on emulsion rheology was studied.
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
1932-1058/2014/8(2)/024104/15/$30.00
8, 024104-1
C 2014 AIP Publishing LLC V
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Biomicrofluidics 8, 024104 (2014)
They found that a decrease in shear rate associated with larger volume fraction of droplets led to more aggregation and less deformation. This yielded an increase in the relative viscosity of emulsions.4 Marangoni effects on drop deformation in extensional flow using surface equations of state for insoluble surfactants, which account for surface saturation were probed. The Marangoni stresses result in smaller surfactant concentration gradient and smaller deformations than the uniform surface tension case at a given capillary number.5 To precisely control the micro-scale droplets of the emulsion and to deliver stable and efficient functions for those micro-scale droplets, more active and accurate control methods are needed. To solve this problem, a few methods applying external forces such as electronic, magnetic, acoustic, and optical sources have been introduced.12–14 Of these external force sources, an optical source that uses light as the energy source makes the fast analysis available, as it can produce ultra-high speed reaction. The light source with high resolution to control the microdroplet can be achieved by using the objective lens, which makes it another important strength of optical force. Furthermore, when using this light as the external force, there is no physical contact between target and source, which means that the non-destructive control is available. Even though a heat effect due to light might exist, its influence is rather small and negligible.15 Because of these advantages, many researchers are employing an optical source as an external source to manipulate nano/micro-scale substances.16–18 Since a single-beam gradient-force optical trap using a laser was reported,19,20 many applications have been introduced, such as particle manipulation, sorting, and analysis.21 Optical tweezers are used to manipulate biological cells22–24 because of the ability to measure the properties of living cells, such as elasticity, stiffness and contractility.25,26 Optical tweezers provide the ability to probe mechanical properties, interactions, and the structure of polymer and colloidal materials at nano/micro-scales. Some researchers coupled optical tweezers with cavityenhanced Raman scattering (CERS) and controlled the size of a single aerosol droplet with nanometer accuracy.17 The deformation shape of a droplet with ultralow interfacial tension using multiple optical trapping forces was controlled.27 In this paper, we simulated the optical force on a surfactant-covered droplet interface to actively control a single droplet deformation and the local distribution of surfactant concentration. In this process, the non-linear relationship between optical forces and single droplet properties was analyzed. The mechanism of the movement of surfactant molecules which has been difficult to be observed in experiments was also studied. Through the analysis of the effects of optical forces on single droplet, we establish a fundamental theory and judge the potential for extending to the application of optical forces on an emulsion system. Using this approach, we can actively control the rheological properties of emulsions and obtain the required surfactant distribution. We used a hybrid model of the lattice Boltzmann method (LBM) coupled with a convective-diffusion equation for surfactant transport, to create a single droplet and employed a ray-optics regime to numerically express the optical forces. Overall contents and objectives of this paper are shown in Fig. 1.
II. NUMERICAL METHOD
A coupled numerical model is needed to develop a surfactant-covered emulsion droplet. The LBM commonly used for micro-scale simulation is coupled with the time-dependent convective-diffusion surfactant model. Optical forces are integrated as an external force term with LBM. Although optical forces are distributed on the interface, these forces are overwhelmingly focused on a small area. Therefore, the summation of all those forces is alternatively applied as a point source.
A. Two component LBM
The isothermal and single-relaxation model is derived from the Boltzmann kinetic equation,
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Biomicrofluidics 8, 024104 (2014)
FIG. 1. Flow chart for summarizing the objectives of the study.
›f þ c $f ¼ Xðf Þ; ›t
(1)
where f is the density distribution function, c is the lattice velocity, and Xðf Þ is the collision term. In LBM, the collision term of Eq. (1) which is very complicated is simplified by the Bhatnagar-Gross-Krook (BGK) approximation for practical calculations. Hence, the simplified equation can be expressed as @fi 1 þ ci rfi ¼ ðfieq fi Þ; @t s
(2)
where feq is the equilibrium distribution function and s is the physical relaxation time. Equation (2) is assumed to be valid along a specified direction. D3Q19 LBM used in our code has nineteen specific velocity direction vectors, with central vector of speed zero, described in Fig. 2. Equation (2) then is discretized as follows: f i ðx þ ci Dt; t þ Dt Þ f i ðx; t Þ ¼
Dt eq ½f ðx; t Þ f i ðr; t Þ: s i
(3)
We call the left side of Eq. (3) “streaming” and the right side “collision.” ci ¼ ei =Dt is the lattice velocity in the ith direction, and the lattice time step, Dt, is unity. The equilibrium distribution function can be obtained by the following formula:
3 9 3 2 feq ¼ qxi 1 þ 2 ci u þ 4 ðci uÞ 2 u u ; c 2c 2c
(4)
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Biomicrofluidics 8, 024104 (2014)
FIG. 2. Velocity vectors for D3Q19 lattice Boltzmann method.
xi ¼ ½1=3; 1=36; 1=36; 1=18; 1=36; 1=36; 1=36; 1=18; 1=36; 1=18; 1=36; 1=36; 1=18; 1=36; 1=36; 1=36; 1=18; 1=36; 1=18 ;
(5)
where c ¼ 1 is the ratio of the lattice space and the lattice time step, xi is the weighting factor that determines the relative probability of a particle movement in the ith direction, and q and u are the macroscopic density and velocity, which are calculated as follows:
q¼
Q1 X
fi ¼
i¼0
qu ¼
Q1 X
fieq ;
(6)
i¼0
ci f i ¼
i¼0
Q1 X
Q1 X
ci fieq :
(7)
i¼0
To simulate a two-immiscible-fluid emulsion, we employed Gunstensen LBM. In the standard Gunstensen model,28 the total distribution function is the summation of the two fluid distribution functions as follows: f i ¼ ri þ b i :
(8)
By defining the interface between two fluids, it is possible to make a phase field, qN ¼
ri bi : ri þ bi
(9)
The collision step is performed by the total distribution function. In order to apply the surface tension force, the method of Lishchuk et al.29 is employed here using the following macroscopic force (F(x)): 1 FðxÞ ¼ akrqN ; 2
(10)
where a is the interfacial tension and k is the droplet curvature. The relationship between the source term in LBM equation and the macroscopic force F(x) is calculated by the Guo et al.30 method:
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Biomicrofluidics 8, 024104 (2014)
1 3ðei u Þ þ 9ðei u Þej Fð xÞ ; /i ðxÞ ¼ xi 1 2s " # 18 1 X 1 fi ei þ FðxÞ ; u ¼ q i¼1 2
(11)
(12)
where u is the corrected velocity used in the calculation of the equilibrium function. The local interfacial tension affects the surface tension force which influences the corrected velocity. Therefore, Marangoni flow induced by the interfacial tension gradient can be applied. Because the red fluid density is 1 and the blue fluid density is 1, the interfacial density is defined as lower than the red fluid density and higher than the blue fluid density. Through this method, the interfacial nodes can be tracked and the surface forces are calculated only in the interfacial nodes. After that, the fluid is separated in two parts, red and blue fluids, using the following relationship:31,32 fiR ðx; t þ dt Þ ¼
qR qR qB ð Þ þ b x; t þ d xi cos hf hi jci j; f t i R B R B q þq q þq
fiB ð x; t þ dt Þ ¼ fi ð x; t þ dt Þ fiR ðx; t þ dt Þ;
(13) (14)
where fiR and fiB are the post-collision and post-segregation distribution functions of the red and blue fluids, and hf and hi are the polar angle of the color field, and the angle of the velocity link, b is the segregation parameter, and fi is the post collision distribution function. After segregation, these two fluids are propagated separately and macroscopic properties are obtained through each distribution functions. The units used in this method are identified as follows: spatial lattice unit [lu], time step [ts], mass l, and lattice mole [lmol] which are equivalent to 100 nm, 4.8 ns, 3:6 1016 g, and 3:2 1014 mol in real physical units. B. The surfactant model
Many studies have sought to achieve surfactant-covered droplet simulation. A hybrid lattice Boltzmann model with surfactant distribution was developed.33 The surfactant transport on the droplet interface is governed by the following time-dependent convective-diffusion equation: @t C þ rs ðus CÞ þ kCun ¼ Ds r2 C;
(15)
where C is the surfactant concentration, us ; un are the surface velocity and the normal velocity, k is the surface curvature, and Ds is the diffusivity which is typically quite small.34 In Eq. (12), @t C signifies the change in the local surfactant concentration as time flows, rs ðus CÞ is the convection term related to interfacial velocity, and kCun shows how surface curvature affects the surfactant concentration. The right term, Ds r2 C, is the diffusive contribution. The hopscotch explicit and unconditionally stable finite difference method is used here to solve the convective-diffusion equation, and then, coupled with Gunstensen LBM. This scheme uses two is calculated at each grid, for consecutive sweeps through the domain, the first sweep Cnþ1 i;j nþ1 is calculated at which i þ j þ n is even, by a simple explicit scheme. The second sweep Ci;j each grid point, for which i þ j þ n is odd, by a simple implicit scheme. More information on the solution of the convective-diffusion equation can be found in Ref. 33. If surfactant molecules adhere to the interface, the surface tension is governed by the Langmuir non-linear equation, r ¼ r0 ð1 þ E0 lnð1 C Þ;
(16)
where r is the local surface tension, r0 is the surface tension for the clean surface without surfactant, E0 ¼ C1r0RT is the surfactant elasticity that determines the sensitivity of the surface
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Biomicrofluidics 8, 024104 (2014)
tension to changes in surfactant concentration, C ¼ CC1 is the relative surfactant concentration, and C1 is the saturation surfactant concentration. Equation (16) is not enough for explaining the relationship between the surfactant concentration and the surface tension because of the local accumulation of surfactant which drives the surface tension to zero. This local accumulation is prevented by two mechanisms in the interfacial stress balance. First, the local curvature rises to increase the local area and dilute the interfacial surfactant concentration. This phenomenon is governed by the Laplace pressure. Second, the Marangoni effect which is the mass transfer along the interface due to the surface tension gradient prevents any further accumulation of surfactant. The Marangoni stress is expressed as follows:3 rs r ¼ @C r rs C:
(17)
The partial derivative term in Eq. (14) is expressed as @C r ¼ RT =ð1C=C1 Þ , indicating that the Marangoni stress increases as a rise of the interfacial gradient of the surfactant concentration.
C. The optical force model
When a single laser beam is focused on a certain point inside of a transparent particle, the attractive force pulling the particle toward the highly focused area of the beam is formed. This process is called optical traps (also optical tweezers). There are two main forces that govern the optical traps, the gradient and scattering force. The gradient force is defined as the force pointing in the direction of the intensity gradient of the light and the scattering force is defined as the force pointing in the direction of the incident light. If the gradient force is dominant over the scattering force, the trapping force is generated. Many attempts have been made to numerically and theoretically express optical forces.35–38 Analytic expressions for radiation force on a sphere in a focused Gaussian beam by the photon stream method in the ray-optics regime were derived.33 The force equations are expressed as follows: sin 2ðh1 h2 Þ þ R sin 2h1 2 rp sin 2h1 cos udh1 du; (18) Iðq; zÞ R sin 2h1 T 1 þ R2 þ 2R cos 2h2 0 0 ð ð n0 2p p=2 2 cos 2ðh1 h2 Þ þ R cos 2h1 2 rp sin 2h1 dh1 du; (19) Fs ¼ Iðq; zÞ 1 þ R cos 2h1 T 2c 0 0 1 þ R2 þ 2R cos 2h2
n0 Fg ¼ 2c
ð 2p ð p=2
2
where Fg and Fs are the gradient and scattering forces, n0 is the medium refractive index, c is the speed of light in free space, Iðq; zÞ is the Gaussian beam intensity profile, R and T are the Fresnel reflectance and transmittance, h1 and u are the incident and polar angle, and rp is the radius of the sphere. The integration was performed using the trapezoidal numerical method. The summation of two forces from Eqs. (18) and (19) is applied on the droplet interface for all cases. At first, we tracked the interface by locating the phase field qN . The incident, reflected and polar angles on each interface nodes were calculated by using normal components of the interface nodes. Then, the gradient and scattering forces were calculated with other given values such as the medium refractive index, the speed of light, the Fresnel reflectance and transmittance, and the radius of the sphere. Through the vector summation of those two forces, the total optical force was finally obtained. This total optical force was applied on the interface as a point source term, with a direction pulling toward the outside of the droplet. A flow chart explaining the algorithm of the hybrid LBM for surfactant-covered droplets and for applying the optical force is shown in Fig. 3. To justify our algorithm in terms of physics, we should verify how optical tweezers work in the real world. For soft matters, such as living cells, two beads are attached on both ends of
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Biomicrofluidics 8, 024104 (2014)
FIG. 3. Flow chart for the hybrid LBM for surfactant-covered droplets and for applying optical forces.
the cell membrane and pulled by optical tweezers. Therefore, the cell is not directly affected by the laser beam. However, because beads cannot be attached at the interface of droplets, the laser beam directly passes through the droplet interface. This can be expressed as a source term on the interface but not a sink term outside of the droplets. III. RESULTS
To evaluate the suitability, we validated the developed model with some experimental results. After that, the deformation and the surfactant distribution changed due to optical forces with diverse droplet properties and flow conditions. A. Model validation
To validate the proposed model, the results of droplet deformation and shape in multiple optical traps were compared with the experimental results.27 In the experiment, heptane droplets in water with surfactant, which have low interfacial tension, were used along with various optical powers (11–27 mW). Multiple optical traps were used to obtain various shapes of the droplet such as ellipse, triangle, and rectangle. We set the same conditions as in the experimental case, in which 2 lm heptane droplets in water were used, with low interfacial tension (0.5). This is because the higher surfactant coverage decreases the surface tension. The CKL number characterizes not only the relationship between the optical force and the interfacial tension but also the drop breakup. When the optical force reached 12 pN, the droplet broke up with low surface tension (
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BIOMICROFLUIDICS 8, 024104 (2014)
Study for optical manipulation of a surfactant-covered droplet using lattice Boltzmann method Se Bin Choi,1 Sasidhar Kondaraju,2 and Joon Sang Lee1,a) 1
Department of Mechanical Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea 2 Department of Mechanical Engineering, Indian Institute of Technology, Block III #174, Main Building, New Delhi 110016, India (Received 10 January 2014; accepted 3 March 2014; published online 12 March 2014)
In this study, we simulated deformation and surfactant distribution on the interface of a surfactant-covered droplet using optical tweezers as an external source. Two optical forces attracted a single droplet from the center to both sides. This resulted in an elliptical shape deformation. The droplet deformation was characterized as the change of the magnitudes of surface tension and optical force. In this process, a non-linear relationship among deformation, surface tension, and optical forces was observed. The change in the local surfactant concentration resulting from the application of optical forces was also analyzed and compared with the concentration of surfactants subjected to an extensional flow. Under the optical force influence, the surfactant molecules were concentrated at the droplet equator, which is totally opposite to the surfactants behavior under extensional flow, where the molecules were concentrated at the poles. Lastly, the quasi-equilibrium surfactant distribution was obtained by combining the effects of the optical forces with the extensional flow. All simulations were executed by the lattice Boltzmann method which is a C 2014 AIP Publishing LLC. powerful tool for solving micro-scale problems. V [http://dx.doi.org/10.1063/1.4868368]
I. INTRODUCTION
Understanding the rheological behavior of emulsion systems has a broad range of industrial applications, in the food, medical, cosmetic, polymer, water purification, and pharmaceutical industries. An emulsion is defined as a mixture of two or more liquids that are immiscible, such as water-in-oil (W/O) or oil-in-water (O/W). A number of products are made up of droplet-based immiscible mixtures (emulsion). Surfactants which adhere to the droplet interface and reduce its surface tension are mainly used to obtain the stability of emulsion and used to control both the droplet size and deformability. Polymer manufacturing industries strive to provide enhanced methods of emulsion control in order to satisfy market demand. The behavior of surfactant-covered droplets in simple flow has been studied experimentally and numerically.1–11 The deformation and burst of small droplets in steady 2-dimensional linear flows were experimentally investigated, comparing the results with predictions of several available asymptotic deformation and burst theories, as well as numerical calculations.1 The effects of surfactants on drop deformation and breakup in extensional flows at low Reynolds number were discussed. The analytical and numerical results indicated that surfactants behave differently at low and high capillary numbers, which leads to different levels of deformation.2 The drop breakup mechanism in extensional flow caused by the transport of surfactant molecules on the interface was numerically simulated, which results in changes in local surface tension.3 The influence of deformation and de-aggregation of droplets on emulsion rheology was studied.
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
1932-1058/2014/8(2)/024104/15/$30.00
8, 024104-1
C 2014 AIP Publishing LLC V
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024104-2
Choi, Kondaraju, and Lee
Biomicrofluidics 8, 024104 (2014)
They found that a decrease in shear rate associated with larger volume fraction of droplets led to more aggregation and less deformation. This yielded an increase in the relative viscosity of emulsions.4 Marangoni effects on drop deformation in extensional flow using surface equations of state for insoluble surfactants, which account for surface saturation were probed. The Marangoni stresses result in smaller surfactant concentration gradient and smaller deformations than the uniform surface tension case at a given capillary number.5 To precisely control the micro-scale droplets of the emulsion and to deliver stable and efficient functions for those micro-scale droplets, more active and accurate control methods are needed. To solve this problem, a few methods applying external forces such as electronic, magnetic, acoustic, and optical sources have been introduced.12–14 Of these external force sources, an optical source that uses light as the energy source makes the fast analysis available, as it can produce ultra-high speed reaction. The light source with high resolution to control the microdroplet can be achieved by using the objective lens, which makes it another important strength of optical force. Furthermore, when using this light as the external force, there is no physical contact between target and source, which means that the non-destructive control is available. Even though a heat effect due to light might exist, its influence is rather small and negligible.15 Because of these advantages, many researchers are employing an optical source as an external source to manipulate nano/micro-scale substances.16–18 Since a single-beam gradient-force optical trap using a laser was reported,19,20 many applications have been introduced, such as particle manipulation, sorting, and analysis.21 Optical tweezers are used to manipulate biological cells22–24 because of the ability to measure the properties of living cells, such as elasticity, stiffness and contractility.25,26 Optical tweezers provide the ability to probe mechanical properties, interactions, and the structure of polymer and colloidal materials at nano/micro-scales. Some researchers coupled optical tweezers with cavityenhanced Raman scattering (CERS) and controlled the size of a single aerosol droplet with nanometer accuracy.17 The deformation shape of a droplet with ultralow interfacial tension using multiple optical trapping forces was controlled.27 In this paper, we simulated the optical force on a surfactant-covered droplet interface to actively control a single droplet deformation and the local distribution of surfactant concentration. In this process, the non-linear relationship between optical forces and single droplet properties was analyzed. The mechanism of the movement of surfactant molecules which has been difficult to be observed in experiments was also studied. Through the analysis of the effects of optical forces on single droplet, we establish a fundamental theory and judge the potential for extending to the application of optical forces on an emulsion system. Using this approach, we can actively control the rheological properties of emulsions and obtain the required surfactant distribution. We used a hybrid model of the lattice Boltzmann method (LBM) coupled with a convective-diffusion equation for surfactant transport, to create a single droplet and employed a ray-optics regime to numerically express the optical forces. Overall contents and objectives of this paper are shown in Fig. 1.
II. NUMERICAL METHOD
A coupled numerical model is needed to develop a surfactant-covered emulsion droplet. The LBM commonly used for micro-scale simulation is coupled with the time-dependent convective-diffusion surfactant model. Optical forces are integrated as an external force term with LBM. Although optical forces are distributed on the interface, these forces are overwhelmingly focused on a small area. Therefore, the summation of all those forces is alternatively applied as a point source.
A. Two component LBM
The isothermal and single-relaxation model is derived from the Boltzmann kinetic equation,
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.42.202.150 On: Mon, 24 Nov 2014 17:14:50
024104-3
Choi, Kondaraju, and Lee
Biomicrofluidics 8, 024104 (2014)
FIG. 1. Flow chart for summarizing the objectives of the study.
›f þ c $f ¼ Xðf Þ; ›t
(1)
where f is the density distribution function, c is the lattice velocity, and Xðf Þ is the collision term. In LBM, the collision term of Eq. (1) which is very complicated is simplified by the Bhatnagar-Gross-Krook (BGK) approximation for practical calculations. Hence, the simplified equation can be expressed as @fi 1 þ ci rfi ¼ ðfieq fi Þ; @t s
(2)
where feq is the equilibrium distribution function and s is the physical relaxation time. Equation (2) is assumed to be valid along a specified direction. D3Q19 LBM used in our code has nineteen specific velocity direction vectors, with central vector of speed zero, described in Fig. 2. Equation (2) then is discretized as follows: f i ðx þ ci Dt; t þ Dt Þ f i ðx; t Þ ¼
Dt eq ½f ðx; t Þ f i ðr; t Þ: s i
(3)
We call the left side of Eq. (3) “streaming” and the right side “collision.” ci ¼ ei =Dt is the lattice velocity in the ith direction, and the lattice time step, Dt, is unity. The equilibrium distribution function can be obtained by the following formula:
3 9 3 2 feq ¼ qxi 1 þ 2 ci u þ 4 ðci uÞ 2 u u ; c 2c 2c
(4)
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024104-4
Choi, Kondaraju, and Lee
Biomicrofluidics 8, 024104 (2014)
FIG. 2. Velocity vectors for D3Q19 lattice Boltzmann method.
xi ¼ ½1=3; 1=36; 1=36; 1=18; 1=36; 1=36; 1=36; 1=18; 1=36; 1=18; 1=36; 1=36; 1=18; 1=36; 1=36; 1=36; 1=18; 1=36; 1=18 ;
(5)
where c ¼ 1 is the ratio of the lattice space and the lattice time step, xi is the weighting factor that determines the relative probability of a particle movement in the ith direction, and q and u are the macroscopic density and velocity, which are calculated as follows:
q¼
Q1 X
fi ¼
i¼0
qu ¼
Q1 X
fieq ;
(6)
i¼0
ci f i ¼
i¼0
Q1 X
Q1 X
ci fieq :
(7)
i¼0
To simulate a two-immiscible-fluid emulsion, we employed Gunstensen LBM. In the standard Gunstensen model,28 the total distribution function is the summation of the two fluid distribution functions as follows: f i ¼ ri þ b i :
(8)
By defining the interface between two fluids, it is possible to make a phase field, qN ¼
ri bi : ri þ bi
(9)
The collision step is performed by the total distribution function. In order to apply the surface tension force, the method of Lishchuk et al.29 is employed here using the following macroscopic force (F(x)): 1 FðxÞ ¼ akrqN ; 2
(10)
where a is the interfacial tension and k is the droplet curvature. The relationship between the source term in LBM equation and the macroscopic force F(x) is calculated by the Guo et al.30 method:
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Choi, Kondaraju, and Lee
Biomicrofluidics 8, 024104 (2014)
1 3ðei u Þ þ 9ðei u Þej Fð xÞ ; /i ðxÞ ¼ xi 1 2s " # 18 1 X 1 fi ei þ FðxÞ ; u ¼ q i¼1 2
(11)
(12)
where u is the corrected velocity used in the calculation of the equilibrium function. The local interfacial tension affects the surface tension force which influences the corrected velocity. Therefore, Marangoni flow induced by the interfacial tension gradient can be applied. Because the red fluid density is 1 and the blue fluid density is 1, the interfacial density is defined as lower than the red fluid density and higher than the blue fluid density. Through this method, the interfacial nodes can be tracked and the surface forces are calculated only in the interfacial nodes. After that, the fluid is separated in two parts, red and blue fluids, using the following relationship:31,32 fiR ðx; t þ dt Þ ¼
qR qR qB ð Þ þ b x; t þ d xi cos hf hi jci j; f t i R B R B q þq q þq
fiB ð x; t þ dt Þ ¼ fi ð x; t þ dt Þ fiR ðx; t þ dt Þ;
(13) (14)
where fiR and fiB are the post-collision and post-segregation distribution functions of the red and blue fluids, and hf and hi are the polar angle of the color field, and the angle of the velocity link, b is the segregation parameter, and fi is the post collision distribution function. After segregation, these two fluids are propagated separately and macroscopic properties are obtained through each distribution functions. The units used in this method are identified as follows: spatial lattice unit [lu], time step [ts], mass l, and lattice mole [lmol] which are equivalent to 100 nm, 4.8 ns, 3:6 1016 g, and 3:2 1014 mol in real physical units. B. The surfactant model
Many studies have sought to achieve surfactant-covered droplet simulation. A hybrid lattice Boltzmann model with surfactant distribution was developed.33 The surfactant transport on the droplet interface is governed by the following time-dependent convective-diffusion equation: @t C þ rs ðus CÞ þ kCun ¼ Ds r2 C;
(15)
where C is the surfactant concentration, us ; un are the surface velocity and the normal velocity, k is the surface curvature, and Ds is the diffusivity which is typically quite small.34 In Eq. (12), @t C signifies the change in the local surfactant concentration as time flows, rs ðus CÞ is the convection term related to interfacial velocity, and kCun shows how surface curvature affects the surfactant concentration. The right term, Ds r2 C, is the diffusive contribution. The hopscotch explicit and unconditionally stable finite difference method is used here to solve the convective-diffusion equation, and then, coupled with Gunstensen LBM. This scheme uses two is calculated at each grid, for consecutive sweeps through the domain, the first sweep Cnþ1 i;j nþ1 is calculated at which i þ j þ n is even, by a simple explicit scheme. The second sweep Ci;j each grid point, for which i þ j þ n is odd, by a simple implicit scheme. More information on the solution of the convective-diffusion equation can be found in Ref. 33. If surfactant molecules adhere to the interface, the surface tension is governed by the Langmuir non-linear equation, r ¼ r0 ð1 þ E0 lnð1 C Þ;
(16)
where r is the local surface tension, r0 is the surface tension for the clean surface without surfactant, E0 ¼ C1r0RT is the surfactant elasticity that determines the sensitivity of the surface
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Choi, Kondaraju, and Lee
Biomicrofluidics 8, 024104 (2014)
tension to changes in surfactant concentration, C ¼ CC1 is the relative surfactant concentration, and C1 is the saturation surfactant concentration. Equation (16) is not enough for explaining the relationship between the surfactant concentration and the surface tension because of the local accumulation of surfactant which drives the surface tension to zero. This local accumulation is prevented by two mechanisms in the interfacial stress balance. First, the local curvature rises to increase the local area and dilute the interfacial surfactant concentration. This phenomenon is governed by the Laplace pressure. Second, the Marangoni effect which is the mass transfer along the interface due to the surface tension gradient prevents any further accumulation of surfactant. The Marangoni stress is expressed as follows:3 rs r ¼ @C r rs C:
(17)
The partial derivative term in Eq. (14) is expressed as @C r ¼ RT =ð1C=C1 Þ , indicating that the Marangoni stress increases as a rise of the interfacial gradient of the surfactant concentration.
C. The optical force model
When a single laser beam is focused on a certain point inside of a transparent particle, the attractive force pulling the particle toward the highly focused area of the beam is formed. This process is called optical traps (also optical tweezers). There are two main forces that govern the optical traps, the gradient and scattering force. The gradient force is defined as the force pointing in the direction of the intensity gradient of the light and the scattering force is defined as the force pointing in the direction of the incident light. If the gradient force is dominant over the scattering force, the trapping force is generated. Many attempts have been made to numerically and theoretically express optical forces.35–38 Analytic expressions for radiation force on a sphere in a focused Gaussian beam by the photon stream method in the ray-optics regime were derived.33 The force equations are expressed as follows: sin 2ðh1 h2 Þ þ R sin 2h1 2 rp sin 2h1 cos udh1 du; (18) Iðq; zÞ R sin 2h1 T 1 þ R2 þ 2R cos 2h2 0 0 ð ð n0 2p p=2 2 cos 2ðh1 h2 Þ þ R cos 2h1 2 rp sin 2h1 dh1 du; (19) Fs ¼ Iðq; zÞ 1 þ R cos 2h1 T 2c 0 0 1 þ R2 þ 2R cos 2h2
n0 Fg ¼ 2c
ð 2p ð p=2
2
where Fg and Fs are the gradient and scattering forces, n0 is the medium refractive index, c is the speed of light in free space, Iðq; zÞ is the Gaussian beam intensity profile, R and T are the Fresnel reflectance and transmittance, h1 and u are the incident and polar angle, and rp is the radius of the sphere. The integration was performed using the trapezoidal numerical method. The summation of two forces from Eqs. (18) and (19) is applied on the droplet interface for all cases. At first, we tracked the interface by locating the phase field qN . The incident, reflected and polar angles on each interface nodes were calculated by using normal components of the interface nodes. Then, the gradient and scattering forces were calculated with other given values such as the medium refractive index, the speed of light, the Fresnel reflectance and transmittance, and the radius of the sphere. Through the vector summation of those two forces, the total optical force was finally obtained. This total optical force was applied on the interface as a point source term, with a direction pulling toward the outside of the droplet. A flow chart explaining the algorithm of the hybrid LBM for surfactant-covered droplets and for applying the optical force is shown in Fig. 3. To justify our algorithm in terms of physics, we should verify how optical tweezers work in the real world. For soft matters, such as living cells, two beads are attached on both ends of
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Choi, Kondaraju, and Lee
Biomicrofluidics 8, 024104 (2014)
FIG. 3. Flow chart for the hybrid LBM for surfactant-covered droplets and for applying optical forces.
the cell membrane and pulled by optical tweezers. Therefore, the cell is not directly affected by the laser beam. However, because beads cannot be attached at the interface of droplets, the laser beam directly passes through the droplet interface. This can be expressed as a source term on the interface but not a sink term outside of the droplets. III. RESULTS
To evaluate the suitability, we validated the developed model with some experimental results. After that, the deformation and the surfactant distribution changed due to optical forces with diverse droplet properties and flow conditions. A. Model validation
To validate the proposed model, the results of droplet deformation and shape in multiple optical traps were compared with the experimental results.27 In the experiment, heptane droplets in water with surfactant, which have low interfacial tension, were used along with various optical powers (11–27 mW). Multiple optical traps were used to obtain various shapes of the droplet such as ellipse, triangle, and rectangle. We set the same conditions as in the experimental case, in which 2 lm heptane droplets in water were used, with low interfacial tension (0.5). This is because the higher surfactant coverage decreases the surface tension. The CKL number characterizes not only the relationship between the optical force and the interfacial tension but also the drop breakup. When the optical force reached 12 pN, the droplet broke up with low surface tension (
