## Theoretical analysis for the optical deformation of emulsion droplets |

Optics Express, Vol. 22, Issue 4, pp. 4523-4538 (2014)

http://dx.doi.org/10.1364/OE.22.004523

Acrobat PDF (6569 KB)

### Abstract

We propose a theoretical framework to predict the three-dimensional shapes of optically deformed micron-sized emulsion droplets with ultra-low interfacial tension. The resulting shape and size of the droplet arises out of a balance between the interfacial tension and optical forces. Using an approximation of the laser field as a Gaussian beam, working within the Rayleigh-Gans regime and assuming isotropic surface energy at the oil-water interface, we numerically solve the resulting shape equations to elucidate the three-dimensional droplet geometry. We obtain a plethora of shapes as a function of the number of optical tweezers, their laser powers and positions, surface tension, initial droplet size and geometry. Experimentally, two-dimensional droplet silhouettes have been imaged from above, but their full side-on view has not been observed and reported for current optical configurations. This experimental limitation points to ambiguity in differentiating between droplets having the same two-dimensional projection but with disparate three-dimensional shapes. Our model elucidates and quantifies this difference for the first time. We also provide a dimensionless number that indicates the shape transformation (ellipsoidal to dumbbell) at a value ≈ 1.0, obtained by balancing interfacial tension and laser forces, substantiated using a data collapse.

© 2014 Optical Society of America

## 1. Introduction

1. A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. **2006**, 4515–4517 (2006). [CrossRef]

2. D. A. Woods, C. D. Mellor, J. M. Taylor, C. D. Bain, and A. D. Ward, “Nanofluidic networks created and controlled by light,” Soft Matter **7**, 2517–2520 (2011). [CrossRef]

3. R. Karlsson, A. Karlsson, A. Ewing, P. Dommersnes, J.-F. Joanny, A. Jesorka, and O. Orwar, “Chemical analysis in nanoscale surfactant networks,” Anal. Chem. **78**, 5961–5968 (2006). [CrossRef] [PubMed]

4. G. Hirasaki, C. Miller, and M. Puerto, “Recent advances in surfactant EOR,” SPE J. **16**, 889–907 (2011) [CrossRef]

5. P. J. H. Bronkhorst, G. J. Streekstra, J. Grinbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. **69**, 1666–1673 (1995). [CrossRef] [PubMed]

8. J. Dharmadhikari, S. Roy, A. Dharmadhikari, S. Sharma, and D. Mathur, “Torque-generating malaria-infected red blood cells in an optical trap,” Opt. Express **12**, 1179–1184 (2004). [CrossRef] [PubMed]

9. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

12. D. McGloin, “Optical tweezers: 20 years on,” Philos. Trans. R. Soc. A **364**, 3521–3537 (2006). [CrossRef]

13. S. Block, L. Goldstein, and B. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature **348**, 348–352 (1990). [CrossRef] [PubMed]

15. R. J. Davenport, G. J. Wuite, R. Landick, and C. Bustamante, “Single-molecule study of transcriptional pausing and arrest by E. coli RNA polymerase,” Science **287**, 2497–2500 (2000). [CrossRef] [PubMed]

16. A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. **30**, 139–142 (1973). [CrossRef]

17. A. Casner and J.-P. Delville, “Giant deformations of a liquid-liquid interface induced by the optical radiation pressure,” Phys. Rev. Lett. **87**, 054503 (2001). [CrossRef] [PubMed]

*et al.*showed that the interfacial tension of heptane droplets could be lowered to

*γ*≈ 10

^{−5}− 10

^{−6}

*Nm*

^{−1}by the addition of suitably selected surfactants [1

1. A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. **2006**, 4515–4517 (2006). [CrossRef]

*μm*) water droplets using pulsed laser beams [18

18. J.-Z. Zhang and R. K. Chang, “Shape distortion of a single water droplet by laser-induced electrostriction,” Opt. Lett. **13**, 916–918 (1988). [CrossRef] [PubMed]

19. E. Evans and A. Yeung, “Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration,” Biophys. J. **56**, 151–160 (1989). [CrossRef] [PubMed]

*κ*is the bending modulus and

*ξ*is a length scale based on the radius of curvature of the droplet surface, in the hydrodynamic limit the interfacial tension contribution dominates over the curvature energy contribution.

## 2. Mathematical model for droplet deformation

*R*(

*θ*,

*ϕ*) in spherical polar coordinates, where

*R*defines the distance between the interface and a preassigned fixed origin. In the absence of any external forces, a liquid droplet assumes a spherical shape, as a result of two antagonistic forces: the interfacial tension which tends to minimise the area, and the bulk pressure of the internal fluid which translates into a force acting along the local normal to an infinitesimal surface element

*dS*. The energy function for an isolated droplet can be written as: Thus, the internal pressure for an isolated droplet at equilibrium turns out to be

**r**on an infinitesimal surface element

*dS*at the oil-water interface can be written as: where

*P*is the optical pressure due to momentum transfer from the laser field to the interface,

_{opt}*P*is the Laplace pressure (which by convention acts inwards in the case of a convex droplet surface), and

_{lap}*P*is the internal pressure within the droplet, which we will treat as an unknown variable dependent on the specific experimental configuration, but uniform throughout the droplet since we are only considering equilibrium structures.

_{int}*P*(

_{lap}**r**) acting on a surface element on the oil-water interface (interfacial tension

*γ*), at a position

**r**and with outward normal

**n̂**(

**r**), is calculated using the Young-Laplace equation: from using Gauss’ theorem on the first integral in Eq. (1).

*P*(

_{opt}**r**) across the oil-water interface is calculated by considering the momentum transferred to the interface as light is reflected and refracted at the interface. An exact solution for the light field in the presence of a dielectric object could be calculated in the form of a series expansion within the framework of T-matrix theory [25

25. M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles: a tutorial review,” J. Quantum Spectrosc. Radiat. Transfer **110**, 808–832 (2009). [CrossRef]

26. F. Xu, J. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A **79**, 053808 (2009). [CrossRef]

*R*= 2

_{d}*μm*.

**r**and with outward normal

**n̂**(

**r**), then for an incident beam with momentum density

*p*

_{0}

*n*

_{1}

**ŝ**, where

**ŝ**is the direction of the Poynting vector, we can apply standard laws of reflection and refraction [28] to determine the directions

**ŝ**

*and*

_{t}**ŝ**

*of the transmitted and reflected rays (see Chapter 1 of [28]). In our case we are specifically interested in the momentum density transferred to the interface, In this particular case, the momentum transfer is normal to the surface and the resultant expression can be considerably simplified. After some algebraic manipulation the optical pressure acting on the surface (in the direction of the outward normal) is determined to be: where*

_{r}*μ*=

**ŝ**·

**n̂**,

*F*and

_{r}*F*are the Fresnel power reflection and transmission coefficients for the angle of incidence

_{t}*θ*= arccos(|

_{inc}*μ*|), and

*n*

_{1}and

*n*

_{2}are the refractive indices for the media in which the incoming and refracted beams are propagating.

*p*

_{0}

*n*

_{1}

**ŝ**of the beam. The fact that our droplets are several wavelengths in diameter, significantly larger than the trapping beam waist, implies that a series expansion around the beam focus, such as that presented in [29

29. J. P. Barton and D. R. Alexander, “Fifthorder corrected electromagnetic field components for a fundamental Gaussian beam,” Appl. Phys. **66**, 2800–2802 (1989). [CrossRef]

26. F. Xu, J. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A **79**, 053808 (2009). [CrossRef]

*θ*,

*ϕ*) on the droplet interface can be written in terms of its rate of normal motion: and hence where the (

**n̂**·

**r̂**) term can be understood as taking into account the fact that the interface normal is not collinear to the (fixed) radial direction along which

*R*(

*θ*,

*ϕ*) is measured – see Fig. 1, and hence

*R*will in general vary faster than the rate of normal motion of the interface.

*t*’ does not represent real “time”. Therefore, in the absence of a full hydrodynamic description of the enclosed fluid the dynamical equations do not capture transient shapes that can be compared to experiments. The parameter

*β*in Eq. (7) represents the computational convergence rate for our numerical procedure. By leaving

*β*as a free parameter without physical meaning, we are able to calculate the steady-state shapes that are the focus of this work in as rapidly and computationally efficiently way as possible. A higher value of

*β*would correspond to rapid convergence to the equilibrium configuration. However,

*β*cannot be chosen to be arbitrarily large as it compromises the stability of the numerical scheme used to integrate Eq. (7).

*P*representing the internal pressure of the droplet:

_{int}*R*.

_{d}## 3. Numerical implementation

*N*×

_{θ}*N*nodes equally spaced in (

_{ϕ}*θ*,

*ϕ*) space. Node (i,j) lies at coordinates (

*i*<

*N*and 0 ≤

_{θ}*j*<

*N*. For each node a radius

_{ϕ}*R*is defined, each representing a point (

_{ij}*R*,

_{ij}*θ*,

_{i}*ϕ*) on the surface of the droplet (the oil-water interface), so that the radius of each node defines a “spoke” extending from the origin in the direction (

_{j}*θ*,

*ϕ*) – see Fig. 1. We then solve the equation of motion (Eq. (7)) on this discretized grid, using finite-difference expressions as an estimate for local derivatives on the surface of the droplet.

**n̂**as follows [31]: where

*r*is the radial coordinate and

*R*(

*θ*,

*ϕ*) is the radius of a “spoke” at a given (

*θ*,

*ϕ*) coordinate.

*N*= 25 and

_{θ}*N*= 24 nodes. Once the droplet shape has converged using this coarse mesh, we refine to

_{ϕ}*N*= 51 and

_{θ}*N*= 48. Figure 2 shows a comparison between the converged shapes of a droplet with

_{ϕ}*R*= 5.0

_{d}*μm*in an increasing number of optical traps. It can be seen that the refinement improves the shapes of the droplets in the regions of high curvature where the optical traps are positioned, whilst only minor improvements can be observed for the rest of the droplet. A mesh size of

*N*= 13 and

_{θ}*N*= 12 is also shown in green in Fig. 2. Such a coarse mesh is incapable of ensuring a smooth surface is obtained, while the initial mesh size of

_{ϕ}*N*= 25 and

_{θ}*N*= 24 nodes represents a good compromise between computational demands and accuracy, being able to give a reasonable initial estimate of the droplet shape that can later be refined. Furthermore, the calculation of the Laplace pressure would be erroneous with an insufficient number of mesh points. Our choice of

_{ϕ}*N*and

_{θ}*N*is also verified by considering the well-known fact that the integral of the mean curvature

_{ϕ}*H*over a closed surface

*S*is equal to zero [32

32. D. Blackmore and L. Ting, “Surface integral of its mean curvature vector,” SIAM Rev. **27**, 569–572 (1985) [CrossRef]

*N*and

_{θ}*N*nodes is increased in our model, this calculated quantity will tend to zero.

_{ϕ}## 4. Results and discussion

*λ*= 1064

*nm*have been modelled for results reported in this paper. Measured values of the refractive index of a heptane droplet [33

33. M. Goffredi, V. T. Liveri, and G. J. Vassallo, “Refractive index of water-AOT-n-heptane microemulsions,” J. Solut. Chem. **22**, 941–949 (1993). [CrossRef]

*n*

_{1}= 1.38 and

*n*

_{2}= 1.33 respectively, have been used in our calculations, unless otherwise stated.

### 4.1. Single optical trap

*R*= 2.0

_{d}*μm*, and a surface tension

*γ*= 10

^{−6}

*Nm*

^{−1}, as a function of increasing laser power

*P*

_{0}. The radius of the beam waist for each optical trap was kept constant at

*w*

_{0}= 0.282

*μm*, which corresponds to a numerical aperture of

*NA*= 1.20. As the laser power increases the droplet elongates to assume a lozenge form, with its long axis parallel to the direction of propagation of light (taken to be along the +

*z*axis). For laser powers

*P*

_{0}≳ 0.055

*W*the droplet has a dumbbell-like shape with an hour-glass connecting two spherical caps. For these configurations one of the principal curvatures is negative. Experimental observations of these deformations have so far been limited to their two-dimensional projections along the axis of laser propagation [1

1. A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. **2006**, 4515–4517 (2006). [CrossRef]

*z*. If we consider the gradient force alone then we would expect a symmetric shape, reflecting the symmetry of the intensity distribution in the trapping laser field. However the scattering force (which acts in the local direction of propagation of the laser field) breaks this symmetry and pushes the interface, and hence the droplet, in the direction of propagation of the laser and leads to “bulging” of the droplet beyond the laser focus.

*μm*

^{−1}corresponding to that of a perfect sphere with radius

*R*= 2.0

_{d}*μm*, except at the “tips” of the droplet. As the laser power is increased we see a larger variation in the mean curvature, with the lowest mean curvature at the waist of the droplet, and the highest mean curvature at the two spherical caps. For laser powers greater than

*P*

_{0}= 0.12

*W*the mean curvature at the waist becomes negative.

*xy*projections, the deformation of these droplets in a single optical trap is only known due to a decrease in the maximum projected radius [1

**2006**, 4515–4517 (2006). [CrossRef]

*R*as a function of the laser power

_{xy}*P*

_{0}for a constant value of interfacial tension

*γ*= 10

^{−6}

*Nm*

^{−1}and initial droplet size

*R*= 5.0

_{d}*μm*for different values of numerical aperture

*NA*is shown in Fig. 4(a).

*R*changes non-monotonically as a function of

_{xy}*P*

_{0}, with the minimum occurring at the transition to a dumbbell-like form, a transition that we will discuss in more detail below. The initial linear decrease in the slope of the

*R*vs.

_{xy}*P*

_{0}curve is due to the elongation along the

*z*-axis. Beyond the transition point, the equilibrated configurations show a narrowing of the neck region connecting the two spherical caps. A similar trend is observed for different values of the initial droplet size

*R*as shown in Fig. 4(b). Figures 4(c) and 4(d) show the similar qualitative trends for a droplet with

_{d}*R*= 5.0

_{d}*μm*for different values of the interfacial tension 10

^{−7}≲

*γ*≲ 10

^{−6}

*Nm*

^{−1}and refractive index of a droplet

*n*

_{1}respectively.

*R*) and the interfacial tension (

_{d}*γ*), and inversely proportional to the refractive index ratio

*ñ*. The relationship between the laser power and the numerical aperture is slightly more complicated. We find that the laser power

*P*

_{0}∝ exp(

*NA*). These relationships allow us to define a dimensionless number which characterises when a droplet will deform into a dumbbell-like shape, for a given initial radius, surface tension, refractive index ratio and numerical aperture: When

*N*≳ 1.0 the droplet deforms into a dumbbell-like shape, whereas for

_{d}*N*≲ 1.0 the droplet only elongates in the direction of the propagation of light. Figure 5(a) shows this for a constant initial spherical radius

_{d}*R*= 5.0

_{d}*μm*whilst varying the numerical aperture for the optical tweezer. Figure 5(b) shows the variation of

*R*vs.

_{xy}*N*for different starting radii, whilst using a numerical aperture of

_{d}*NA*= 1.10. To obtain a data collapse for the graphs in Fig. 5(b) we rescale the observed

*R*values as: Figure 5(c) also shows the same trend for a variation in the refractive index of the droplet

_{xy}*n*

_{1}. As it can be seen, the minimum in Figs. 5(a) and 5(c) and maximum in Fig. 5(b), indicating the transition to a dumbbell-like shape, occurs at

*N*≈ 1.0.

_{d}*N*≲ 0.7 the interfacial tension energy is stronger than that of the optical tweezer, resulting in a linear deformation of the droplet. Between

_{d}*N*≈ 0.7 and 1.0 the forces exerted by the laser field begin to overcome the interfacial tension of the droplet, until

_{d}*N*≈ 1.0 at which the optical forces dominate. The droplet interface then conforms locally to the iso-intensity contours of the laser beam, which results in the observed dumbbell-like geometry.

_{d}### 4.2. Multiple optical traps

*μm*in the appropriate directions, and we ensure that the droplet shape has converged at each stage before the lasers are moved again.

*P*=

_{total}*P*

_{0}and we obtain a value of

*N*= 2.18 for the given

_{d}*R*,

_{d}*γ*and

*NA*. Hence this is beyond the transition to a dumbbell-like shape, as shown in Figs. 6(a), 7(a) and 8(a). The hour-glass connecting the two spherical caps has a negative mean curvature. As each individual laser is then moved outwards from the centre, the surface of the droplet at the “tips” or “corners” retains this concave shape. This observation is interesting since from just viewing the two-dimensional projections one might assume the surfaces to be convex along the axis of propagation of light.

*R*= 5.0

_{d}*μm*and interfacial tension

*γ*= 10

^{−6}

*Nm*

^{−1}, deformed in four optical traps each with a laser power of

*P*

_{0}= 0.1

*W*and numerical aperture

*NA*= 0.8 has negative mean curvature on the top and bottom faces, in addition to that seen on the side faces in Fig. 8. This indicates that the deformation of droplets by optical forces is extremely sensitive to the selected parameters.

*et al.*[1

**2006**, 4515–4517 (2006). [CrossRef]

*R*= 2.5

_{d}*μm*was taken for the initial spherical geometry of the droplet, and an interfacial tension of

*γ*≈ 10

^{−6}

*Nm*

^{−1}was reported. For the optical traps, each laser had a numerical aperture of

*NA*= 1.20 and a total combined laser power of

*P*= 24

_{total}*mW*was equally distributed between the total number of lasers. The experimentally observed shapes and steady-state shapes predicted by our model are presented in the first and second rows of Fig. 9 respectively, for an increasing number of optical traps.

*et al.*we obtain droplets with conical ends. The high electric field at the laser foci exerts a very strong force on the interface, and when combined with volume conservation and surface curvature considerations, this results in a locally very small radius of curvature at the tips of the shape, as previously modelled by Stone

*et al.*[34

34. H. A. Stone, J. R. Lister, and M. P. Brenner, “Drops with conical ends in electric and magnetic fields,” Proc. R. Soc. London A **455**, 329–347 (1999). [CrossRef]

35. G. Taylor, “Disintegration of water drops in an electric field,” Proc. R. Soc. London A **280**, 383–397 (1964). [CrossRef]

*N*= 0.35 deduced from the parameters reported by Ward

_{d}*et al.*[1

**2006**, 4515–4517 (2006). [CrossRef]

*xy*projections of the deformed droplets and those obtained experimentally. We hope that this work will prompt experimental studies to visualise the three-dimensional configurations as a function of the physical parameters explored in this paper.

## 5. Conclusion

*γ*. The optical traps were described using a far-field scalar model of a tightly focused Gaussian beam, within the Rayleigh-Gans regime.

*et al.*for similar parameter values. The close agreement between theoretical predictions of two-dimensional projections in the

*xy*plane and experiments for known geometries and parameter values gives us confidence in the correctness of our predicted three-dimensional droplet shapes. It is worth noting that these predictions of three-dimensional droplet shapes for large deformations (where the linear response breaks down) have not been reported in the literature. Experiments are currently under development to measure three-dimensional shapes for comparison with our model

*N*, obtained using dimensional arguments balancing antagonistic surface tension and optical forces. The mathematical expression has been substantiated using a data collapse of the numerical solution of the shape equations.

_{d}## Acknowledgments

## References and links

1. | A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. |

2. | D. A. Woods, C. D. Mellor, J. M. Taylor, C. D. Bain, and A. D. Ward, “Nanofluidic networks created and controlled by light,” Soft Matter |

3. | R. Karlsson, A. Karlsson, A. Ewing, P. Dommersnes, J.-F. Joanny, A. Jesorka, and O. Orwar, “Chemical analysis in nanoscale surfactant networks,” Anal. Chem. |

4. | G. Hirasaki, C. Miller, and M. Puerto, “Recent advances in surfactant EOR,” SPE J. |

5. | P. J. H. Bronkhorst, G. J. Streekstra, J. Grinbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. |

6. | J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Lett. |

7. | J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. |

8. | J. Dharmadhikari, S. Roy, A. Dharmadhikari, S. Sharma, and D. Mathur, “Torque-generating malaria-infected red blood cells in an optical trap,” Opt. Express |

9. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

10. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

11. | P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of Gold nanoparticles,” Nano Lett. |

12. | D. McGloin, “Optical tweezers: 20 years on,” Philos. Trans. R. Soc. A |

13. | S. Block, L. Goldstein, and B. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature |

14. | G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express |

15. | R. J. Davenport, G. J. Wuite, R. Landick, and C. Bustamante, “Single-molecule study of transcriptional pausing and arrest by E. coli RNA polymerase,” Science |

16. | A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. |

17. | A. Casner and J.-P. Delville, “Giant deformations of a liquid-liquid interface induced by the optical radiation pressure,” Phys. Rev. Lett. |

18. | J.-Z. Zhang and R. K. Chang, “Shape distortion of a single water droplet by laser-induced electrostriction,” Opt. Lett. |

19. | E. Evans and A. Yeung, “Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration,” Biophys. J. |

20. | H. M. Lai, P. T. Leung, K. L. Poon, and K. Young, “Electrostrictive distortion of a micrometer-sized droplet by a laser pulse,” J. Opt. Soc. Am. B |

21. | I. Brevik and R. Kluge, “Oscillations of a water droplet illuminated by a linearly polarized laser pulse,” J. Opt. Soc. Am. B |

22. | H. Chrabi, D. Lasseux, E. Arquis, R. Wunenburger, and J.-P. Delville, “Stretching and squeezing of sessile dielectric drops by the optical radiation pressure,” Phys. Rev. E |

23. | P. C. F. Møller and L. B. Oddershede, “Quantification of droplet deformation by electromagnetic trapping,” Europhys. Lett. |

24. | S. Å. Ellingsen, “Theory of microdroplet and microbubble deformation by Gaussian laser beam,” J. Opt. Soc. Am. B |

25. | M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles: a tutorial review,” J. Quantum Spectrosc. Radiat. Transfer |

26. | F. Xu, J. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A |

27. | H. C. van de Hulst, |

28. | M. Herzberger, |

29. | J. P. Barton and D. R. Alexander, “Fifthorder corrected electromagnetic field components for a fundamental Gaussian beam,” Appl. Phys. |

30. | A. E. Siegman, |

31. | M. Spivak, |

32. | D. Blackmore and L. Ting, “Surface integral of its mean curvature vector,” SIAM Rev. |

33. | M. Goffredi, V. T. Liveri, and G. J. Vassallo, “Refractive index of water-AOT-n-heptane microemulsions,” J. Solut. Chem. |

34. | H. A. Stone, J. R. Lister, and M. P. Brenner, “Drops with conical ends in electric and magnetic fields,” Proc. R. Soc. London A |

35. | G. Taylor, “Disintegration of water drops in an electric field,” Proc. R. Soc. London A |

**OCIS Codes**

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

(240.0240) Optics at surfaces : Optics at surfaces

(240.6700) Optics at surfaces : Surfaces

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: November 1, 2013

Revised Manuscript: January 8, 2014

Manuscript Accepted: January 19, 2014

Published: February 20, 2014

**Virtual Issues**

Vol. 9, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

David Tapp, Jonathan M. Taylor, Alex S. Lubansky, Colin D. Bain, and Buddhapriya Chakrabarti, "Theoretical analysis for the optical deformation of emulsion droplets," Opt. Express **22**, 4523-4538 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4523

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### References

- A. D. Ward, M. G. Berry, C. D. Mellor, C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. 2006, 4515–4517 (2006). [CrossRef]
- D. A. Woods, C. D. Mellor, J. M. Taylor, C. D. Bain, A. D. Ward, “Nanofluidic networks created and controlled by light,” Soft Matter 7, 2517–2520 (2011). [CrossRef]
- R. Karlsson, A. Karlsson, A. Ewing, P. Dommersnes, J.-F. Joanny, A. Jesorka, O. Orwar, “Chemical analysis in nanoscale surfactant networks,” Anal. Chem. 78, 5961–5968 (2006). [CrossRef] [PubMed]
- G. Hirasaki, C. Miller, M. Puerto, “Recent advances in surfactant EOR,” SPE J. 16, 889–907 (2011) [CrossRef]
- P. J. H. Bronkhorst, G. J. Streekstra, J. Grinbergen, E. J. Nijhof, J. J. Sixma, G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69, 1666–1673 (1995). [CrossRef] [PubMed]
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