Study for optical manipulation of a surfactant-covered droplet using lattice Boltzmann method

Abstract

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 powerful tool for solving micro-scale problems.

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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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. 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, 13, 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 micro-droplet 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, 17, 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 cells^{22, 23, 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 cavity-enhanced 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.

Figure 1.

Flow chart for summarizing the objectives of the study.

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.

Two component LBM

The isothermal and single-relaxation model is derived from the Boltzmann kinetic equation,

$$\frac{\mathit{\partial }\mathit{f}}{\mathit{\partial }\mathit{t}}+\mathit{c}\cdot \mathit{\nabla }\mathit{f}=\mathbf{\Omega }(\mathit{f}),$$ (1)

where f is the density distribution function, c is the lattice velocity, and $\Omega (f)$ 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

$$\frac{\partial f_i}{\partial t}+c_i\cdot \nabla f_i=\frac{1}{\tau }(f_i^{\mathit{e}\mathit{q}}-f_i),$$ (2)

where $f_{\mathit{e}\mathit{q}}$ is the equilibrium distribution function and τ 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.

Figure 2.

Velocity vectors for D3Q19 lattice Boltzmann method.

Equation 2 then is discretized as follows:

$${\mathit{f}}_{\mathit{i}}\left(\mathit{x}+{\mathit{c}}_{\mathit{i}}\Delta \mathit{t},\mathit{t}+\Delta \mathit{t}\right)-{\mathit{f}}_{\mathit{i}}\left(\mathit{x},\mathit{t}\right)=\frac{\Delta \mathit{t}}{\mathit{\tau }}[{\mathit{f}}_{\mathit{i}}^{\mathit{eq}}\left(\mathit{x},\mathit{t}\right)-{\mathit{f}}_{\mathit{i}}\left(\mathit{r},\mathit{t}\right)].$$ (3)

We call the left side of Eq. 3 “streaming” and the right side “collision.” $c_i=e_i/\Delta t$ is the lattice velocity in the $i\text{th}$ direction, and the lattice time step, $\Delta \mathrm{t}$, is unity. The equilibrium distribution function can be obtained by the following formula:

$$f_{\mathit{e}\mathit{q}}=\rho {\omega }_i\left[1+\frac{3}{c^2}{c}_i\cdot u+\frac{9}{2c^4}{\left(c_i\cdot u\right)}^2-\frac{3}{2c^2}u\cdot u\right],$$ (4)

$$\begin{array}{c}{\omega }_i=[1/3;1/36;1/36;1/18;1/36;1/36;1/36;1/18;1/36;\\ 1/18;1/36;1/36;1/18;1/36;1/36;1/36;1/18;1/36;1/18] ,\end{array}$$ (5)

where c = 1 is the ratio of the lattice space and the lattice time step, ${\omega }_i$ is the weighting factor that determines the relative probability of a particle movement in the $i\text{th}$ direction, and ρ and u are the macroscopic density and velocity, which are calculated as follows:

$$\rho ={\displaystyle \sum }_{i=0}^{Q-1}{f}_i={\displaystyle \sum }_{i=0}^{Q-1}{f}_i^{\mathit{e}\mathit{q}},$$ (6)

$$\rho \mathrm{u}={\displaystyle \sum }_{i=0}^{Q-1}{c}_i{f}_i={\displaystyle \sum }_{i=0}^{Q-1}{c}_i{f}_i^{\mathit{e}\mathit{q}}.$$ (7)

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:

$$f_i=r_i+b_i.$$ (8)

By defining the interface between two fluids, it is possible to make a phase field,

$${\rho }^N=\frac{r_i-b_i}{r_i+b_i}.$$ (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)):

$$\mathrm{F}\left(\mathrm{x}\right)= -\frac{1}{2}\alpha k\nabla {\rho }^N,$$ (10)

where α 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:

$${\varphi }_{\mathrm{i}}\left(x\right)={\omega }_i\left(1-\frac{1}{2\tau }\right)\left[3\left({\mathrm{e}}_{\mathrm{i}}-u^{*}\right)+9\left(e_i{u}^{*}\right)e_j\right]F\left(x\right) ,$$ (11)

$${\mathrm{u}}^{*}=\frac{1}{\rho }\left[{\displaystyle \sum }_{i=1}^{18}{f}_i{e}_i+\frac{1}{2}F(x)\right],$$ (12)

where ${\mathrm{u}}^{*}$ 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}

$${\overline{\overline{f}}}_i^{\mathrm{R}}\left(x,\hspace{0.17em}t+{\delta }_t\right)=\hspace{0.17em}\frac{{\rho }^{\mathrm{R}}}{{\rho }^{\mathrm{R}}+{\rho }^B}{\overline{f}}_i\left(x,\hspace{0.17em}t+{\delta }_t\right)+\beta \frac{{\rho }^R{\rho }^B}{{\rho }^{\mathrm{R}}+{\rho }^{\mathrm{B}}}{\omega }_i\cos \left({\theta }_f-{\theta }_i\right)|c_i|,$$ (13)

$$\hspace{0.17em}{\overline{\overline{f}}}_i^{\mathrm{B}}\left(x,\hspace{0.17em}t+{\delta }_t\right)={\overline{f}}_i\left(x,\hspace{0.17em}t+{\delta }_t\right)-{\overline{\overline{f}}}_i^{\mathrm{R}}\left(x,\hspace{0.17em}t+{\delta }_t\right),$$ (14)

where ${\overline{\overline{f}}}_i^{\mathrm{R}}$ and ${\overline{\overline{f}}}_i^{\mathrm{B}}$ are the post-collision and post-segregation distribution functions of the red and blue fluids, and ${\theta }_f$ and ${\theta }_i$ are the polar angle of the color field, and the angle of the velocity link, β is the segregation parameter, and ${\overline{f}}_i$ 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 μ, and lattice mole [lmol] which are equivalent to 100 nm, 4.8 ns, $3.6\times {10}^{-16}$ g, and $3.2\times {10}^{-14}$ mol in real physical units.

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:

$${\partial }_t\Gamma +{\nabla }_s\cdot \left(u_s\Gamma \right)+k\Gamma u_n=D_s{\nabla }^2\Gamma ,$$ (15)

where Γ is the surfactant concentration, $u_s,\hspace{0.17em}{u}_n$ are the surface velocity and the normal velocity, k is the surface curvature, and $D_s$ is the diffusivity which is typically quite small.³⁴ In Eq. 12, ${\partial }_t\Gamma$ signifies the change in the local surfactant concentration as time flows, ${\nabla }_s\cdot \left(u_s\Gamma \right)$ is the convection term related to interfacial velocity, and $k\Gamma u_n$ shows how surface curvature affects the surfactant concentration. The right term, $D_s{\nabla }^2\Gamma$, 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 consecutive sweeps through the domain, the first sweep ${\Gamma }_{i,j}^{n+1}$ is calculated at each grid, for which i + j + n is even, by a simple explicit scheme. The second sweep ${\Gamma }_{i,j}^{n+1}$ is calculated at each grid point, for which i + j + 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,

$$\sigma ={\sigma }_0(1+E_0\ln \left(1-{\Gamma }^{*}\right),$$ (16)

where σ is the local surface tension, ${\sigma }_0$ is the surface tension for the clean surface without surfactant, $E_0=\frac{{\Gamma }_{\infty }RT}{{\sigma }_0}$ is the surfactant elasticity that determines the sensitivity of the surface tension to changes in surfactant concentration, ${\Gamma }^{*}=\frac{\Gamma }{{\Gamma }_{\infty }}$ is the relative surfactant concentration, and ${\Gamma }_{\infty }$ 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

$$-{\nabla }_s\sigma =-{\partial }_{\Gamma }\sigma \cdot {\nabla }_s\Gamma .$$ (17)

The partial derivative term in Eq. 14 is expressed as ${\partial }_{\Gamma }\sigma =-\raisebox{1ex}{$\mathit{R}\mathit{T}$}\!\left/ \!\raisebox{-1ex}{$(1-\Gamma /{\Gamma }_{\infty })$}\right.$, indicating that the Marangoni stress increases as a rise of the interfacial gradient of the surfactant concentration.

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, 36, 37, 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:

$$F_g=-\frac{n_0}{2c}{{\displaystyle \int }}_0^{2\pi }{{\displaystyle \int }}_0^{\pi /2}I(\rho ,z)\left[R\sin 2{\theta }_1-T^2\frac{\sin 2\left({\theta }_1-{\theta }_2\right)+R\sin 2{\theta }_1}{1+R^2+2R\cos 2{\theta }_2}\right]r_p^2\sin 2{\theta }_1\cos \phi d{\theta }_{1}d\phi ,$$ (18)

$$F_s=\frac{n_0}{2c}{{\displaystyle \int }}_0^{2\pi }{{\displaystyle \int }}_0^{\pi /2}I(\rho ,z)\left[1+R\cos 2{\theta }_1-T^2\frac{\cos 2\left({\theta }_1-{\theta }_2\right)+R\cos 2{\theta }_1}{1+R^2+2R\cos 2{\theta }_2}\right]r_p^2\sin 2{\theta }_{1}d{\theta }_{1}d\phi ,$$ (19)

where $F_g$ and $F_s$ are the gradient and scattering forces, $n_0$ is the medium refractive index, c is the speed of light in free space, $I(\rho ,\hspace{0.17em}z)$ is the Gaussian beam intensity profile, R and T are the Fresnel reflectance and transmittance, ${\theta }_1$ and φ are the incident and polar angle, and $r_p$ is the radius of the sphere. The integration was performed using the trapezoidal numerical method. The summation of two forces from Eqs. 18, 19 is applied on the droplet interface for all cases. At first, we tracked the interface by locating the phase field ${\rho }^N$. 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.

Figure 3.

Flow chart for the hybrid LBM for surfactant-covered droplets and for applying optical forces.

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 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.

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.

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 $\mu \mathrm{m}$ heptane droplets in water were used, with low interfacial tension (<$3\times {10}^{-6}\mathrm{N}/\mathrm{m})$ and subjected to 24 mW optical power, which is equivalent to 8 pN force. In our code, the droplet radius and the interfacial tension were 20 lu and 0.0001 $\hspace{0.17em}{\text{mu}/\text{ts}}^2$($\cong (1.5\times {10}^{-6}\hspace{0.17em}\mathrm{N}/\mathrm{m})\times {(4.8\frac{\text{ns}}{\text{ts}}\hspace{0.17em})}^2÷(3.6\times {10}^{-16}\frac{\mathrm{g}}{\text{mu}})$), respectively. A source term of 0.00512 $\hspace{0.17em}\text{mu}\cdot {\text{lu}/\text{ts}}^2$ ($\cong (8\times {10}^{-12}\mathrm{N})\times {\left(4.8\frac{\text{ns}}{\text{ts}}\right)}^2÷(3.6\times {10}^{-16}\frac{\mathrm{g}}{\text{mu}})÷(100\frac{\text{nm}}{\text{lu}})$) was used to equate to an 8 pN physical force. The segregation parameter β in Eq. 10 which affects the thickness of the interface was set to 0.69 for the stable simulation.³² The code was run until steady state was reached when the normal velocities along the interface approached zero. We defined the deformation index as $\frac{b}{a}$ for the ellipse case, where a is the conjugate radius and b is the transverse radius. For all cases, we employed the characteristic length $4\mathrm{A}/\mathrm{P}$, where A is the cross sectional area of the center and P is the circumference.

The comparison between the experiment and numerical simulation is shown in Fig. 4. In particular, for the ellipse case, the deformation indices of the droplet are 1.53 in the experiment and 1.56 in the numerical simulation, which accounts for a 1.96% difference. For the triangle and the rectangle cases, the proportions of the characteristic length from the experiment versus those from the simulation are 3.0:2.8 and 4.4:4.4, respectively.

Figure 4.

Comparison for validation between (a) experimental cases and (c) 3D numerical cases, and (d) graph for comparison of characteristic length in triangle and rectangle cases. (b) The location and the direction of the optical forces.

The effect of optical forces on droplet behavior

Deformation

To study the relationship between optical forces and droplet properties, we simulated several cases that have 2 optical traps causing elliptic deformation. The deformation as the change of the optical forces, the surface tension and surfactant coverage in still flow was plotted. The domain size was $101\times 101\times 101{\hspace{0.17em}\text{lu}}^3$ and the range of optical force was from 4 pN to 12 pN (equivalent from 0.00256 $\hspace{0.17em}\text{mu}\cdot {\text{lu}/\text{ts}}^2$ to 0.00768 $\hspace{0.17em}\text{mu}\cdot {\text{lu}/\text{ts}}^2$). The non-dimensional surface tension that is for clean surface without surfactant was varied from 0.0001 to 0.001. The surfactant coverage $c_{\mathit{i}\mathit{n}}=\frac{{\Gamma }_i}{{\Gamma }_{\infty }}$ is from 0.17 to 0.5, where ${\Gamma }_i$ and ${\Gamma }_{\infty }$ are the initial and saturated surfactant concentration. The surfactant elasticity is 0.2. The droplet deformation is characterized by $\frac{R_i}{R_0}$, where $R_i$ and $R_0$ are the transverse and original radii of the droplet. The code was run until steady state is reached.

It is clear that when the surface tension decreases, droplet deformation increases under the same optical force, as shown in Fig. 5. However, the deformation increases non-linearly as the surface tension decreases under optical forces higher than 8 pN. The non-linearity becomes stronger for surface tension values below $3\times {10}^{-4}$ (equivalent to 3333 for the inversed surface tension, $1/\sigma$). The non-linearity is not present under 8 pN optical force for the range of the inverse surface tension used in these simulations.

Figure 5.

Deformation-reversed surface tension curves in various optical forces.

In this case, there are two dominant factors that influence the deformation of the droplet, such as optical forces and interfacial tension. At first, the optical force as an attractive force on both sides deforms the droplet into an elliptic shape, and simultaneously, the interfacial tension of the droplet resists deformation, which makes the droplet sustain its original shape. However, as the droplet is deforming by the optical forces, the surfactant concentration is accumulated in the equator (more details in Sec. 3B2). As the deformation increases, the maximum surfactant concentration at the equator of the droplet rises, which means the minimum interfacial tension is reduced. The reduced interfacial tension stimulates the droplet to deforms easily and the non-linearity becomes stronger under the condition that the deformation is large.^{39, 40, 41, 42} To numerically characterize this mechanism, we propose a dimensionless number—Choi-Kondaraju-Lee (CKL) as the ratio of optical force to initial surface force as expressed below,

$$\text{CKL}=\frac{Optical\hspace{0.17em}forces}{Perimeter\times Surface\hspace{0.17em}tension}=\frac{n_{0}Pr}{c{\pi }^2{\omega }_0^2{\sigma }_0},$$ (20)

where P is the power of the laser, ${\omega }_0$ is the beam waist, ${\sigma }_0$ is the initial surface tension, and r is the radius of the droplet. When the CKL number is below 0.5, the relative difference between the optical force and the restoring force is small, which causes the deformation to increase linearly. However, above 0.5, the difference becomes bigger and the optical force overwhelms the restoring force, resulting in a non-linear increase in the deformation with respect to the CKL number, as shown in Fig. 6.

Figure 6.

Graph representing the deformation of the droplet with respect to CKL number defined by the ratio of optical force and surface force.

Figure 6 also shows that, as the surfactant coverage increased, the deformation slightly increased in all ranges of the CKL number and the level of non-linearity becomes stronger in the high CKL number region (>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 (<$1.2\times {10}^{-4}$ equivalent to 8333 for the inversed surface tension) as in Fig. 5. In terms of the CKL number, if the number exceeds 1.2 at 0.17 surfactant coverage ($c_{\mathit{i}\mathit{n}}=\frac{{\Gamma }_i}{{\Gamma }_{\infty }})$, drop breakup will occur. Therefore, the critical CKL number is defined as the last value for which a stable shape could be 1.2 with 0.17 surfactant coverage.

Local distribution of surfactant concentration by optical forces

The local distribution of surfactant concentration, shortly surfactant distribution, is one of the key factors which govern the level of deformation and breakup, because the existence of surfactant molecules tends to decrease the local surface tension. In other words, the local surface tension is directly connected with the surfactant distribution.

We visualized the surfactant distribution on the drop interface when a droplet was deformed by 8 pN optical forces. In this case, the domain size was $123\times 123\times 123{\hspace{0.17em}\text{lu}}^3$, the surfactant elasticity was 0.2 and surface tension was 0.0001.

In Fig. 7a, it shows surfactant distribution on the droplet interface when t = 23.5 μs. Figure 7a indicates that surfactant molecules are concentrated on the equatorial area of the droplet under the influence of the optical forces. In contrast, surfactant concentration in the pole area is lower than the initial surfactant concentration, which means the surfactant molecules are swept from the pole to the equator of the droplet by the interfacial velocity. Surfactant concentration in the equator area did not exceed the saturated concentration due to the Marangoni stresses and surfactant concentration gradient was reduced. Figure 7b shows that the direction of interfacial velocity is toward the equator from both poles. The optical forces attract droplet poles from the center of the droplet and then the surrounding fluid moves to the equator along the interface as a reaction, which leads surfactant molecules to the equator (Fig. 7c). In this process, the Marangoni stresses which are created by surfactant concentration gradient prevent the excess accumulation of surfactant molecules on the droplet poles.

Figure 7.

The droplet deformed by optical forces (a) visualization of the local distribution of surfactant concentration on the droplet interface when t = 23.5 μs, showing that the equator area in z-axis has a higher surfactant concentration than the pole areas in z-axis. (b) Visualization of the interfacial velocity between the equator and the poles. (c) Mechanism of the movement of the surfactant molecules by optical forces. (d) The velocity field around the droplet induced by optical forces.

Figure 8 represents the dimensionless surfactant concentration in the yz-plane with respect to the z coordinate normalized by the original radius of the droplet when t = 23.5 μs. The shape of the graph is convex, which means that surfactant concentration is higher at the equator and lower at the poles. As the surfactant coverage increases from 0.1 to 0.5, the surfactant concentration on overall surface area increases, along with the increase of difference between the maximum and the minimum surfactant concentration.

Figure 8.

Graph representing the dimensionless surfactant concentration in the yz plane as a function of the horizontal coordinate normalized by the droplet radius when optical forces are applied.

Local distribution of surfactant concentration by extensional flow

The mechanism of droplet deformation and breakup by extensional flow has been studied extensively.^{1, 2, 3, 5} We simulated extensional flow on the droplet and compared with the optical force. To simulate uniaxial extensional flow in a domain 123 × 123 × 123 ${\text{lu}}^3$ with a central droplet, von Neumann boundary condition is used. The flow comes in from x- and y-direction and comes out to z-direction. The velocity of outflow is twice relatively to that of inflow to follow mass conservation. The shear rate was set to $\dot{\gamma }=1.23\times {10}^{-6}\hspace{0.17em}{\text{ts}}^{-1}$ and the velocity field at the boundary is defined as follows:

$$u^{\infty }=\dot{\gamma }\left(\begin{array}{ccc}-0.5& 0& 0\\ 0& -0.5& 0\\ 0& 0& 1\end{array}\right)\cdot \mathit{x}\mathit{.}$$ (21)

In this case, the optical force was set to zero, while all other fluid properties and the domain size were kept unchanged. Some cases to show how a clean droplet deforms under extensional flow were simulated and validated with the experimental and numerical results.⁴² In the reference paper,⁴² the viscosity of the continuous phase was set to ${\mu }_c=$ 400 ± 5 mPa·s, viscosity ratio of the dispersed droplet phase and the continuous phase was λ = 0.335, and the interfacial tension of a clean droplet was ${\sigma }_0=$31.2 $\pm \hspace{0.17em}1\hspace{0.17em}\text{mN}/\mathrm{m}$. The deformation changes as an increase in capillary number which is defined as ${\text{Ca}}_0=R{\mu }_c\dot{\gamma }/{\sigma }_0$ have been observed and validated as shown in Fig. 9.

Figure 9.

Comparison of the present model results with the experimental and numerical results of Feigl et al., for the deformation of a clean droplet in extensional flow as a function of the capillary number.

The surfactant molecules were distributed in a completely opposite way as to the optical force case, as shown in Fig. 10a. This is because in extensional flow, the surrounding fluid moves from the equator to the droplet poles, which leads the surfactant molecules to the poles (Figs. 10b, 10c).

Figure 10.

The droplet deformed by extensional flow. (a) Visualization of the local distribution of surfactant concentration on the droplet interface when t = 23.5 μs, showing that the equator area has a lower surfactant concentration than the pole areas. (b) Visualization of the interfacial velocity between the poles and the equator. (c) Mechanism of the movement of the surfactant molecules by extensional flow. (d) the velocity field around the droplet induced by extensional flow.

Figure 11 shows a concave shape, which implies higher surfactant concentration at the poles and lower surfactant concentration at the equator. The difference between the maximum and minimum surfactant concentration becomes larger as the surfactant coverage increases from 0.1 to 0.5.

Figure 11.

Graph representing the dimensionless surfactant concentration in the yz plane as a function of the horizontal coordinate normalized by the droplet radius when extensional flows are applied.

Local distribution of surfactant concentration under both optical forces and extensional flow

As we have obtained totally opposite results from optical forces and extensional flow in terms of the surfactant distribution, it could be assumed that if both are applied at the same time, a specific surfactant distribution could be satisfied. If the forces due to extensional flow are more dominant than the optical forces, the surfactant molecules will gather at the poles. Similarly, if the optical forces are dominant, the surfactant molecules will travel to the equator of the droplet. Therefore, we can numerically compare the effect of the extensional flow with the effect of the optical forces by the following proposed dimensionless number, optical forces-shearing force (OPS):

$$OPS=\frac{optical\hspace{0.17em}forces}{shearing\hspace{0.17em}force}=\frac{2n_{0}P{r}^2}{\pi c{\omega }_0^2\mu \dot{\gamma }A},$$ (22)

where μ is the dynamic viscosity, ${\omega }_0$ is the waist of the focused laser beam, and A is the cross-sectional area of the domain. The shearing force induced by extensional flow is calculated by the formula of μγ · A. Because the dynamic viscosity and the cross-sectional area are fixed, the shearing force depends on the shear rate. Therefore, OPS number can be determined by controlling the shear rate and a quasi-equilibrium surfactant distribution could be realized, as shown in Fig. 12.

Figure 12.

Graph representing the dimensionless surfactant concentration in the yz plane as a function of the horizontal coordinate normalized by the droplet radius in various non-dimensional numbers. At 0.36, quasi-equilibrium was obtained.

How the surfactant concentration is distributed, is an important mechanism for droplets deformation and breakup. In uniaxial extensional flow, a parent droplet breaks up into daughter droplets which have radii 2 orders of magnitude smaller than their parent droplets and significantly reduced surface tensions, which results in droplets with various surface tension values.³ This phenomenon occurs due to tip streaming, which causes that daughter droplets of a much smaller scale are ejected from thin thread formed at the poles of the droplet because of the pressure imbalance. The pressure imbalance is induced by reduced interfacial tension caused by high surfactant concentration at the poles. However, if optical forces are added and the quasi-equilibrium surfactant distribution is realized, a droplet breaks into 2 parts that have the same sizes and the same surface tensions like the clean droplet. As a result, we can create an emulsion system that consists of the same sized droplets, as shown in Fig. 13.

Figure 13.

Droplet breakup with surface tension contours in 3D domain. The surface tension on the interface kept constant during the deformation and the droplet was separated into two daughter droplets with the same size.

CONCLUSIONS

The effects of optical forces on a surfactant-covered droplet were analyzed. Simulation results show that, in still flow, deformation of a single droplet is non-linearly related to surface tension and optical power due to the non-linear elastic deformation of the droplet. To understand the non-linearity, we defined a dimensionless number, CKL, as the ratio of optical forces and surface force of the droplet. Using this number, the drop breakup was observed to occur over the critical CKL number (1.2) with 0.17 surfactant coverage.

The local surfactant concentration of the deformed droplet by optical forces is higher at the equator than the poles, which is an opposite phenomenon to the deformed droplet by extensional flow. This is because of the directions of the interfacial velocity, which are opposite each other. The surfactant molecules are swept toward the equator in the direction of the interfacial velocity under optical forces, but the mechanism of extensional flow is totally different. Using this phenomenon, even though there has been deformation, the quasi-equilibrium state in terms of the surfactant distribution is realized by combining the optical forces with the extensional flow. This technique will make it possible to achieve a uniform emulsion system composed of droplets of the same size and surface tension.