1. Introduction

Laser trapping and manipulation [1–3] has an increasing number of applications in science and technology for micro-actuation, photo-modification, and characterization of optical properties ofmicro-objects [4–19], sorting and tracking ofmicro-particles [20,21], probing viscosity in micro-environments [22], surface rheology [23], measuring micro-mechanical properties of DNA molecules [24–28], microtubules [29], polymers [30], or red blood cells [31], controlling texture inside liquid crystals [32–35] or gels [36], and lasing [37]. Laser-tweezers have been realized for various spatial laser beam intensity distributions and adopted for different multi-focus and holographic beam geometries [38, 39].

Recently, deposition of molecular assembles by laser tweezers [40] was demonstrated from a 100-µm-thick film of molecular solution spread under a glass slide. A depression on the surface of the solution film was observed around the axis of beam propagation and a rupture of the liquid surface occurred with deposition of the molecular assembly collected by laser tweezers. The molecular alignment of such deposits can be controlled by polarization of the incident light [41], holding promise for nanoscience and technology applications. Assemblies of chemically different nano-materials can be achieved and patterned with micrometer resolution. Tweezer-assisted concentration of molecular solutions could be used for controlled crystallization of proteins once super-saturation is achieved [42].

In this communication we perform a numerical simulation of the light intensity distribution for tight focusing in the thin film combined with an order of magnitude analysis of mechanical transport phenomena induced by laser heating. Finally, a mechanism of cluster assembly and deposition is suggested in which a crucial role is attributed to the coupled action of laser trapping and laser induced Marangoni convection effectively feeding cluster assembly and deposition within the irradiated spot (focus).

2. Light intensity distribution and laser trapping

Three dimensional finite difference time domain (3D FDTD) calculations were carried out using software package FDTD Solutions (Lumerical Inc.). The light intensity distribution is calculated taking into account modification of the refractive index at the focus. We modelled an ellipsoidally shaped refractive index increase and decrease as observed for focal cross sections (Fig. 1). The refractive index increase would correspond to the assembled material and a void models the limiting case of material expulsion from the focus by cavitation and breakdown [43]. This simulation explores the actual light intensity distribution relevant to the reported experiments [40, 41], where the laser trapping beam was tightly focused through the cover glass and was set close to the glass-liquid boundary (see, Fig. 1). Importantly, the thickness (~100 µm) of liquid film was much larger than the focal region and the liquid had a free boundary with air. The surface depression observed experimentally is taken into consideration since it can also affect the focal intensity distribution and facilitate heating of the liquid surface (Fig. 1).

 

Fig. 1. The 3D-FDTD simulations of the light intensity distribution, E ² _y (a,c,e) for different refractive index distributions shown in (b,d,f). The incident field was a y-polarized Gaussian of amplitude (0,1,0) propagating along the x-axis. The half-angle of the focusing cone was θ=30° (corresponds to the NA=0.67 for focusing in water n=1.33). The regions (a,c,e) are marked by corresponding rectangles in (b,d,f).

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Successful laser trapping and deposition of nano-aggregates has recently been demonstrated [40, 41]. Laser trapping of nano-objects depends on the interplay of the gradient force, F_g and the scattering force, F_s, which are responsible for the localization of a nano/micro-object at the focus and pushing axially out of the focus, respectively [44]:

here, 〈S〉 is the time averaged Poynting vector, E is the electrical field strength, σ scattering cross section, n_m the refractive index of the medium, and α is the polarizability of a nanoparticle of refractive index, n_p, and radius, r:

The nano-particles on a molecular level tend to aggregate due to the attractive van der Waals potential which scales as U_vw~1/l ⁶, where l is the separation between nano-objects. Hence, once inside the focal volume the nano-particles tend to collect into globular aggregates which are then more strongly trapped due to increased polarizability/volume α~r ³ (eqns. 1–2). The stiffness of laser tweezers in the focal plane is proportional to the polarizability and the refractive index of the trapped nano-object: $k_{\mathrm{stiff}}=\frac{\partial F_g}{\partial x}~\alpha ~n_p$ [45], here x is the radial coordinate in the focal plane.

Shear interferometry has demonstrated that a strongly localized focal spot heated the liquid surface, followed by rupturing [41]. In the following we discuss the mechanism of this phenomenon taking into account direct heating of the liquid surface by laser tweezers and heating via heat diffusion and convection.

The light intensity distribution, its filamentation, and aberration due to change of refractive index at the focus was modelled in order to assess whether direct light delivery to a free liquid surface is possible. The high light intensity at the surface of the liquid would facilitate evaporation and surface rupture according to the explanation given in ref. [40]. Since a larger axial extent of the focus (mainly due to spherical aberration) corresponds to a smaller numerical aperture we used the value of NA=0.67 for simulation (Fig. 1); at the higher value of NA the effects of axial elongation of the focus were smaller. Figure 1 demonstrates that there is no effective light intensity delivery to the free surface of solvent under tight focusing conditions with NA>0.5 even a few micrometers from the glass-solvent boundary where the laser focus is placed. Hence, there is no direct heating at the liquid surface and changes of refractive index at the focus, either +Δn or -Δn, cannot produce the strong light intensity redistribution and light filamentation necessary to heat the free surface of liquid at 0.1 mm from the focal spot as in the experiments described in ref. [41].

3. Analysis of transport phenomena

Radiation absorption within the thin layer leads to temperature non-uniformity and associated onset of convective flows which are able to change heat and mass transfer and significantly contribute to the formation of the observed structures [41]. The time for onset of temperature non-uniformity is d ²/a≈0.7ms (for d=0.1 mm and a=14×10^-6 m²/s is the thermal diffusivity of water).

After the onset of the initial temperature distribution defined by the heat conductance equation with the heat source given by laser beam absorption, fluid convection starts. The velocity distribution inside the film with small deviations of the density and viscosity is defined by [46]:

coupled with the continuity equation ∇·u=0, where u is the velocity, p is the pressure, ρ is the density, ν is the kinematic viscosity and F is the external force per volume unit.

In the case of the laser induced temperature non-uniformity the fluid motion is driven by two effects: buoyancy and surface tension. First, the volumetric buoyant force, present in eqn. 3, is approximated by the linear term of the Taylor expansion of the density given by:

where β=-ρ ^-1∂ρ/∂T is the coefficient of volumetric temperature expansion, T ₀ is the initial temperature of the liquid and g is the gravitational constant vector.

Second, the surface tension force associated with the gradient of the temperature induces the fluid motion from the surface. In particular, the resulting shear stress exerted along the surface, τ=dσ/dr=σ_TdT/dr, induces the Newton’s fluid momentum flux in the direction normal to the surface, τ=-νρdu_r/dz, generalized by the following boundary condition:

where n is the vector normal to the surface, σ_T=∂σ/∂T (σ is the surface tension), and ∇_‖ is the tangential derivative operator.

The convection develops with time delay defined by l ²/ν, where l is the characteristic size and ν=10^-6m²/s is the kinematic viscosity of water. For buoyancy driven convection induced in the vertical direction this delay corresponds to d ²/ν≈10 ms. For flow developed by the surface tension associated with the temperature gradient along the surface this delay may be significantly higher, i.e. of order of l ² _T/ν where l_T=(1-10)r ₀ is the extent of the temperature gradient along the surface where r ₀ is the laser beam radius. In the case of r ₀≈0.1 mm one has l ² _T/ν≈0.01-1 s.

The onset of fluid motion modifies the temperature distribution and also mass transfer within the heated volume. The stabilization of diffusion transfer occurs with a delay of ≈l ² _T/D=10-1000 s where D≈10^-9 m²/s is the order of magnitude of the diffusion coefficient of inorganic materials in water. Thus, the development of convective transport within a time of r ² ₀/ν≈0.01-1 s is able to enhance significantly mass transfer.

Moreover, this effect can be more significant. In experiments described in refs. [40,41], polymethylmethacrylate (PMMA) solutions were used. The molecular size of PMMA monomers is approximately $R=\frac{\sqrt[3]{m}}{\rho }\approx 11$ nm where m=1.65×10^-21 kg and ρ=1150 kg/m³ are mass of the PMMA molecule and mass density, correspondingly. A rough estimate of the diffusion coefficient of PMMA in solution can be obtained from the approximation for the spherical particle:

where k is the Boltzmann constant, T is the absolute temperature and µ≈10^-3 Pa s is the dynamic viscosity of water. This expression gives for PMMA solution a value of D≈2×10^-11 m²/s and the delay for stabilization time for diffusion ≈ l ² _T/D=500-5×10⁴ s. This suggests that the induced convection developed within l ² _T/ν≈0.01-1 s can play a very significant role.

Let us estimate the order of magnitude of the possible contribution of the convective motion to heat and mass transfer associated with the convective vortices induced by free andMarangoni convection. The rotation direction of fluid inside both convective loops is the same because free convection induces motion of the most heated zone near the center in the vertical direction, continuing in the radial direction where the fluid cools down, whereas the surface tension gradient induces a fluid vortex directed from the center towards the periphery of the heated zone (Fig. 2).

Free convection is known to play a significant role in heat transport whenever Ra>Ra_c0=680 where the value of Rayleigh number is defined by:

The surface tension convection, called Marangoni convection, is known to become significant when Ma>Ma _c0=81. The value of the Marangoni number is defined by:

where l≈r ₀ is the characteristic size of the temperature change along the surface.

Let us estimate these numbers using data for T=20°C: ρ=998 kg/m³, ν=10^-6 m²/s, a=14×10^-6 m²/s, β=1.8×10^-4 1/K, σ_T=1.7×10^-4 N/m K. First, for the experimentally measured temperature rise of ΔT=1.4 K [40, 41] and d=0.1 mm one finds for Ra≈1.75×10^-4. The estimate for l=r ₀=0.1 gives for Marangoni convection Ma≈1.4. Hence, both types of convection do not produce any significant contribution to the heat transfer.

 

Fig. 2. Schematic of laser trapping and deposition of nano-aggregates. The processes shown in panels (c-e) constitute a positive feedback loop which is key for the final deposition of nano-aggregate. See text for detailed discussion.

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However, the comparison of the value of heat (a=14×10^-6 m²/s) and mass diffusion (D=2×10^-11-10^-9 m²/s) coefficients suggests that Marangoni convection produces a very significant impact on the mass transfer inside the heated zone even in the case of D=10^-9 m²/s. That is, taking into account the analogy between heat and mass transport and substituting the value of D for the value of a one finds ${\mathrm{Ma}}_m=\frac{a}{D}\mathrm{Ma}\approx 2\times {10}^4$ for D=10^-9 m²/s, and ${\mathrm{Ma}}_m=\frac{a}{D}\mathrm{Ma}={10}^6$ for D=2×10^-11 m²/s.

Using now an empirical correlation for Marangoni convection mass transfer contribution suggested for the float zone process under optical heating we estimate roughly the factor by which this convection enhances the mass transfer as compared with the molecular diffusion for 10²<Ma<10⁴ [47, 48]:

The extrapolated estimate shows that even in the case of Ma_m=2×10⁴ (D=10^-9 m²/s) this factor reaches D_eff/D≈15, i.e., a very strong enhancement of mass transfer into the irradiated spot takes place. In the case of PMMA Ma_m=10⁶ (D=2×10^-11 m²/s) one finds D_eff/D≈85. This means that during laser trapping the inducedMarangoni convection significantly increases mass transfer within the area over which significant temperature and surface tension gradients exist, and thus increases the number of particles entrapped in the laser focus. The characteristic time for assembling molecules from the laser heated zone will be correspondingly shortened to l ² _T/D_eff. In particular, for D≈10^-9 m²/s and D_eff/D≈15 one finds l ² _T/D_eff≈0.7-70 s whereas for D≈2×10^-11 m²/s and D_eff/D≈85 one obtains l ² _T/D_eff≈6-600 s. This is consistent with the reported prolonged ~1 min incubation time before a concave dimple starts to form on the surface of the liquid layer [40, 41] (see, Fig. 2(d)).

The increase in the density of entrapped particles inside the laser focus provides positive feedback to the temperature increase. In effect, initially the heat source corresponds only to 1.064 µm laser radiation absorption in water defined by α_w≈0.3 cm^-1 [49]. The increase of PMMA concentration N within the laser focus can provide significant contribution to absorption and heat generation inside the laser spot defined by α_PMMA=σ×N, where σ is the PMMA absorption cross-section. This increase in N and related increase in heat generation is followed by rapid temperature correction over a time scale of d ²/a≃0.7ms. This temperature increase finally leads to water evaporation, and increase in the concentration N within the irradiated spot continues until the increased attractive van der Waals potential, U_vw~N ², finally leads to cluster aggregation and deposition on the substrate surface (see, Fig. 2).

In addition to the above mechanism, the film breaking process and cluster deposition is also due to the effect of mechanical destabilization developed with the gradient of surface tension in the radial direction. This destabilization tends to drag the liquid in a radial direction from the heated zone, where surface tension is minimal, similarly to the process of droplet formation from a liquid jet under the effect of a temperature gradient [50].

Recently, convection inside a molten glass induced by a tightly focused laser beam was observed, where a micro-particle was periodically brought to the focal region and was then expelled out due to its lower refractive index (see online video supplement to ref. [17]). Heating of liquids by laser-tweezers facilitates bubbles’ formation and their aggregation at the laser focus [51]. These observations can be explained by the Marangoni convection mass transport mechanism discussed above. Quantitative modeling of the trapping-assembling phenomenon would require numerical solution of the Navier-Stokes equation (eq. 3) for a particular experimental geometry.

4. Conclusions

The optical and hydrodynamic aspects of laser trapping of nano-/micro-particles have been discussed and the experimentally observed results [40, 41] have been qualitatively explained. Marangoni convection is demonstrated to play a significant role in the process of formation of molecular assemblies using laser tweezers, having been found to increase roughly by 1–2 orders of magnitude the mass transfer of dissolved species into the irradiated spot enhancing the assembly and deposition of molecular aggregates. Moreover, the enhanced aggregation of molecules is able to increase the absorption of the incident laser radiation and to increase the local temperature providing thereby positive feedback to the induced convection flow and cluster assembly.

The local temperature increase starts water evaporation and formation of the final deposits inside the irradiated spot. By changing irradiation conditions, initial solute concentration, layer thickness, and the properties of the solvent used, one can control the process, characteristic dimensions, and structure of the produced assemblies. The mechanism discussed here for the particular case of polymer assembly [40, 41] suggests that this mechanism and experimental technique can also be successfully applied to assembling other types of clusters from solutions.

Financial support provided by a Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan No.19360322 is gratefully acknowledged. We would also like to thank Dr. J. Hester from the Australian Nuclear Science and Technology Organization for valuable comments.