Nanoneedle formation via doughnut beam-induced Marangoni effects

Mamoru Tamura; Mamoru Tamura; Mamoru Tamura; Takashige Omatsu; Takashige Omatsu; Takuya Iida; Takuya Iida; Takuya Iida

doi:10.1364/OE.460962

1. Introduction

A laser processing with a high power laser for drilling and cutting is generally employed in a variety of scientific and industrial tasks with high precision, easy and rapid procedures [1]. Recent advances in laser technology have realized structured light beams with orbital and spin angular momentum and unconventional aspects in light-matter interactions have been revealed [2–4]. An intriguing application is employing optical vortices as structured light sources to fabricate helical nanoneedle structures [5–11]. Optical vortices are characterized by their helical wavefronts, and they carry orbital angular momentum (OAM) [12,13]. Also, owing to their on-axis phase singularity, the optical vortices exhibit a ring-shaped intensity spatial form (doughnut beam). Helical needle structures can be fabricated by irradiation of nanosecond pulses on several materials, such as Ta, Au, and Si. However, the physical mechanisms underlying this phenomenon have not been investigated thoroughly, and the clarification of them would be useful for the further improvement of controllability in nano-processing.

The helicity of needle would be attributed to an optical force, which is the driving force of optical tweezers [14,15]. Optical tweezers enable the remote and non-destructive manipulation of small structures with an optical force arising from light-matter interaction. An optical force that attracts targets toward higher-intensity regions (that is a gradient force) is essential for optical trapping of small objects in liquid media. Moreover, the transfer of photon momentum by light scattering and absorption induces another optical force that pushes targets along the direction of light propagation (that is dissipative force). In particular, light field with angular momentum induces torque on objects [12,16]. Spinning and orbital rotation of trapped objects upon irradiation of circularly polarized optical vortices have been reported previously [17–19]. In addition, the relationship between the needle chirality and optical force dependent on the OAM has been discussed in needle fabrication [5,6]. On the other hand, it has not been clarified in detail how the materials were helically transported, and especially how they were pushed up as a needle structure. The previous study suggested that plasma is generated by a high power laser pulse and driven by an optical force [5]. The study with radiation hydrodynamic simulation suggested that the laser-induced shockwave directed the materials towards the dark core of the irradiated optical vortex to shape the microcone, whereas the pulse energy used in the simulation was ten times higher than that of the experiment [11]. Also, there is a report that the surface tension of molten material could contribute to the formation of needle production with simple analysis of fluid velocity, although the dynamic simulation of mass transfer was not carried out [9].

In fact, we cannot ignore the presence of thermally driven motion owing to surface tension of molten material. For instance, in terms of constructing the characteristic nanostructure by a laser processing, laser-induced periodic surface structure (LIPSS) is well known phenomenon, which leads to the polarization-dependent and periodic micro/sub-micron-scale structures on the irradiated target surface [20]. The physical mechanisms underlying this process has been discussed in detail. Among them, in particular, there is a type of LIPSS driven by the Marangoni effect, which induces interface stress owing to surface tension gradient from temperature inhomogeneity. The Marangoni stress is not desired in conventional laser processing, because it generates burrs around drilled holes. However, periodic heating occurs owing to surface mode interference, thereby leading to Marangoni stress and LIPSS [21,22].

From these backgrounds in processing of structure, we consider that thermal effects, for example, the Marangoni stress would contribute to needle production under doughnut beams with ring-shaped spatial intensity form, for example, optical vortex and axially symmetric polarized beams (with radial or azimuthal polarizations) [23,24]. Assuming the irradiation with a doughnut beam as shown in Fig. 1(a), heating due to spatial inhomogeneity could induce the Marangoni stress which acts as a central force to produce the needle structure. In fact, while the angular momentum is not considered in this study, an experimental result in [7] for a silicon showed that a single pulse of doughnut-shaped beam even with zero orbital angular momentum produced the needle with a less chirality. Here, we investigate the needle production possibility and mechanism thoroughly via a numerical study of molten metallic structures at high temperature under a high-power pulsed doughnut beam, focusing on the light-induced nanosecond dynamics of a fluidic continuum, based on the simulation method of laser processing.

Fig. 1. (a) Schematic of needle formation by Marangoni effect under the irradiation of the doughnut beam. (b)(c) Simulation model of cylindrical coordinates with axial symmetry. (d) Intensity distribution of the incident doughnut beam.

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2. Simulation method and model

To evaluate the thermo-fluid dynamics driven by Marangoni effect as shown in Fig. 1(a), we performed the computational fluid dynamics based on the finite volume method with OpenFOAM v2012 [25]. Therein, the solver icoReactingMultiphaseInterFoam was employed with our additional implementation of the laser source of the doughnut beam, the momentum source due to the Marangoni effect, and the heat source due to thermal radiation. Including these parts, the simulation method is as follows.

The governing equations are given as,

(1)$$\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot ({\rho {\mathbf {u}}} )= 0, $$

(2)$$\frac{{\partial \rho {\mathbf {u}}}}{{\partial t}} + \nabla \cdot ({\rho {\mathbf {uu}}} )={-} \nabla p + \nabla \cdot \mu \left( {\nabla {\mathbf {u}} + {{({\nabla {\mathbf {u}}} )}^\textrm{T}} - \frac{2}{3}({\nabla \cdot {\mathbf {u}}} ){\mathbf {I}}} \right) + \rho {\mathbf {g}} + {{\mathbf {S}}_\textrm{P}} + {{\mathbf {F}}_{\textrm{ST}}} + {{\mathbf {F}}_\textrm{M}}, $$

(3)$$\frac{{\partial \rho {c_\textrm{p}}T}}{{\partial t}} + \nabla \cdot ({\rho {\mathbf {u}}{c_\textrm{p}}T} )= \nabla \cdot ({k\nabla T} )+ {Q_{\textrm{opt}}} + {Q_{\textrm{lat}}} + {Q_{\textrm{the}}}, $$

where ρ is the density, u is the velocity, p is the pressure, µ is the viscosity, g is the gravitational acceleration, c_p is the specific heat capacity, T is the temperature, and k is the thermal conductivity. The shape change of the molten material due to the mass transport and phase change is evaluated using the volume of fluid method [26,27]. The simulation region is fulfilled with the three phases. These present ratios in the arbitrary location are described using the scalar field of the volume fractions of α_s, α_l, and α_g of solid, liquid, and gas (α_s + α_l + α_g = 1). For example, α_g = 1 indicates that the gas is only present. When α_g = α_l = 0.5, there are both gas and liquid phases with each half volume of simulation cell, and their interface is present. Based on the volume fraction of the phases, the material parameters are weighted as ρ = ∑_ξρ_ξα_ξ, µ = ∑_ξµ_ξα_ξ, c_p = ∑_ξc_p,ξα_ξ, and k = ∑_ξk_ξα_ξ (ξ = s, l, g). These fraction distributions change upon solving the transport equation:

(4)$$\frac{{\partial {\alpha _\xi }}}{{\partial t}} + \nabla \cdot ({{\alpha_\xi }{\mathbf {u}}} )= \frac{{{\Gamma _\xi }}}{{{\rho _\xi }}}\textrm{, }({\xi = \textrm{s},\textrm{l},\textrm{g}} ), $$

where Γ_ξ is the mass change rate per volume due to the phase change. In this study, we assumed that the phase change occurs from solid to liquid reversibly and from liquid to gas irreversibly. Respectively defining each rate as Γ_s-l and Γ_l-g, Γ_s-l is given by the Lee model [28–30],

(5)$${\Gamma _{\textrm{s - l}}} = {C_\textrm{L}}{\rho _\xi }{\alpha _\xi }\frac{{T - {T_\textrm{m}}}}{{{T_\textrm{m}}}}\textrm{, }\xi = \left\{ {\begin{array}{ll} \textrm{s}&{({{T_\textrm{m}} < T} )}\\ \textrm{l}&{({T < {T_\textrm{m}}} )} \end{array}} \right., $$

where C_L is an arbitrary constant for the phase-change rate, and T_m is the melting temperature. For the phase-change between liquid and gas, based on the study of Hertz, Knudsen, Schrage, and Tanasawa, the following mass flux rate is given [30–34],

(6)$${F_{\textrm{l - g}}} = \frac{{2{C_\textrm{a}}}}{{2 - {C_\textrm{a}}}}\sqrt {\frac{M}{{2\pi RT_\textrm{v}^3}}} {L_\textrm{v}}{\rho _\textrm{g}}({T - {T_\textrm{v}}} ), $$

where C_a is the accommodation coefficient, M is the molecular weight of gas phase, R is the universal gas constant, T_v is the vaporization temperature, and L_v is the specific latent heat of vaporization. Γ_l-g can be evaluated as a product of F_l-g and the interface area, while F_l-g is zero for T < T_v. Using these rates, Γ_ξ is given as Γ_s = −Γ_s-l, Γ_l = Γ_s-l – Γ_l-g, and Γ_g = Γ_l-g. In this study, we ignored the presence of metal vapor; an air was only present as the gas phase. Thus, the evaporation means a removal of metal from simulation region on the spot, along with the latent heat. As a validity of this treatment, the evaporation speed would be slower because Eq. (6) uses the air properties for M and ${\rho _\textrm{g}}(T) = {{{p_{\textrm{amb}}}M} / {RT}}$. We can change it to use the metal property; nevertheless, we did not employ it, because the fast evaporation make simulation instable. Also, the small value of coefficient C_a = 0.1 was employed. We employed this phenomenological evaporation to avoid the excess rise of temperature as much as possible.

Using the inter-phase porosity model by Voller and Prakash [35],

(7)$${{\mathbf {S}}_\textrm{p}} ={-} \frac{{{A_\textrm{p}}\alpha _{\textrm{sol}}^2{\mathbf {u}}}}{{{{({1 - {\alpha_{\textrm{sol}}}} )}^3} + {\delta _\textrm{p}}}}, $$

describes the artificial momentum source to suppress the velocity in the liquid and solid regions, based on the arbitrary constant A_p (δ_p = 10⁻³). F_ST is the surface tension force per volume, given by,

(8)$${{\mathbf {F}}_{\textrm{ST}}} = \sigma \kappa \nabla \alpha, $$

(9)$$\kappa ={-} \nabla \cdot {\mathbf {n}} ={-} \nabla \cdot \frac{{\nabla \alpha }}{{|{\nabla \alpha } |}}, $$

where σ is the surface tension of the molten material, κ is the interface curvature, and n is the normal vector on the interface. Equations (8) and (9) can be used for the two-phase problem, whereas for more phases, $\nabla \alpha$ is replaced by ${\alpha _\textrm{g}}\nabla {\alpha _\textrm{l}} - {\alpha _\textrm{l}}\nabla {\alpha _\textrm{g}}$ to evaluate $\nabla \alpha$ for the liquid-gas interface. Here, F_M is the Marangoni force per volume. From a literature [36], referring a force attributed on the surface tension, we describe the F_M as,

(10)$${{\mathbf {F}}_\textrm{M}} = \gamma ({\nabla T - {\mathbf {n}^{\prime}}({{\mathbf {n}^{\prime}} \cdot \nabla T} )} )|{\nabla \alpha } |\frac{{2({{\rho_\textrm{l}}{\alpha_\textrm{l}} + {\rho_\textrm{g}}{\alpha_\textrm{g}}} )}}{{{\rho _\textrm{l}} + {\rho _\textrm{g}}}}, $$

where, we employ the normal vector to material ${\mathbf {n}^{\prime}} = {{\nabla \alpha ^{\prime}} / {|{\nabla \alpha^{\prime}} |}}$, ($\alpha ^{\prime} = {\alpha _\textrm{l}} + {\alpha _\textrm{s}}$) to evaluate the tangential component of $\nabla T$ on the material interface, while we evaluated $\nabla \alpha$ only for liquid-gas interface because Marangoni force should be present only on the liquid-gas interface. The Marangoni force, which is originally interpreted as the surface stress, is converted to the volume force by $|{\nabla \alpha } |$. In particular, in Eq. (10), the last fraction based on the continuum surface force method [36,37] modifies the force distribution on the liquid–gas interface, and the force acting on the air region is redistributed to the liquid region. The volume heat source Q_opt is evaluated using the discrete transfer radiation model. It injects the rays with the intensity information as mentioned later at each r. Q_opt was calculated from the absorption of ray intensity by the target material according to the Lambert-Beer law. Q_lat is the heat source due to the phase change, given by the product of the mass change rate and latent heat. Q_the is the heat source owing to thermal radiation, given as follows:

(11)$${Q_{\textrm{the}}} ={-} {\varepsilon _{\textrm{em}}}{\sigma _{\textrm{SB}}}({{T^4} - T_{\textrm{amb}}^4} )|{\nabla \alpha^{\prime}} |, $$

where ɛ_em is the emissivity, σ_SB is the Stefan–Boltzmann constant, and T_amb is the ambient temperature.

Figure 1(b) and 1(c) shows our simulation model. To focus on the extrusion behavior of needle production by the thermal effect of the doughnut beam, we employed the two-dimensional simulation system of cylindrical coordinates with axial symmetry, where the contribution of the optical force due to OAM was ignored. The target material was tantalum (Ta). The pioneering work to produce the chiral needle was carried out with the Ta. To investigate a characteristic of needles, some experimental works have been reported. Initially, the regions of z < 0 and z > 0 were filled with solid tantalum and air, respectively. The solid region was irradiated with the Laguerre-Gaussian (LG) beam. The intensity distribution is shown in Fig. 1(d) and expressed by,

(12)$$I(r) = {T_{\textrm{lig}}}\frac{{2P}}{{\pi w_0^2}}\frac{{2{r^2}}}{{w_0^2}}\exp \left( { - \frac{{2{r^2}}}{{w_0^2}}} \right), $$

where P is the laser power and w₀ is the spot radius. Equation (12) was LG mode of the radial index 0, and the azimuthal index +1 or −1. In addition, to consider light reflection on the interface, we multiplied the factor of a light transmittance at the metal interface T_lig = 0.1 in Eq. (12) in advance. The absorption and heat source distribution in the metal was determined by the absorption coefficient α_abs = 0.01 nm⁻¹. These optical parameters of metal were tentative because there was less information on the optical properties in the high-temperature region. From the start of simulation, the laser was injected only for the first 30 ns corresponding to the pulse duration. Also, after stopping the irradiation, the simulation was continued for several hundred nanoseconds. The final time of simulation was changed along with the spot radius: 300, 400, 500, and 600 ns for the w₀ = 5, 6, 7, and 8 µm, respectively. This is because the larger spot radius requires the higher power to melt the material. The time until the injected energy was diffused and decreased was longer with the higher power; thus, the simulation time was also varied to wait for the cooling and solidification of material.

The thermophysical parameters of tantalum were determined based on previously reported values [38–43]. The model parameters are summarized in Table S1 of Supplement 1. The parameters depend on the temperature; however, the values near room temperature or melting point were employed. For the specific heat capacity of the solid, c_p,s, which varies largely depending on the temperature, we employed a value of approximately 2000K. The viscosity, surface tension, and temperature coefficient were adopted from previously reported values [38–40]. In addition, the viscosity and surface tension were also temperature dependent. We employed the values in the high-temperature region, as shown in Supplement 1, Table S1. To investigate the parameter dependence, different values were employed for the viscosity and temperature coefficient of the surface tension. For example, |γ| can be considered as 0.21, 0.25, 0.29 mN/m·K [38–40]. (Note, it is originally valid around the melting temperature). Based on these values, we employed the close values, and a slightly larger value was considered to investigate the parameter dependence, as shown in Table S1.

3. Results and discussion

Figures 2(a)–(f) show the time evolution of the typical simulation results. The corresponding sequential change is exhibited at the upper right one in Visualization 1. In the simulation, the scalar values of α_ξ (ξ = s, l, g for solid, liquid, and gas) were used to express the different states of material based on the volume of fluid method. To draw the different states separately, α_s – α_l was evaluated. Figure 2(g) shows the distribution of the integrated heat source over the z-axis (absorbed intensity of light). As described before, the material was heated by the laser during the first 30 ns. The radial dependence of the heat source reflected the LG mode. Laser heating increased the temperature immediately and melted the irradiated region. The driving force of this formation can be understood from Fig. 2(h). Based on the Marangoni effect, a surface with a lower surface tension is pulled by a higher tension surface. As the surface tension decreases with an increase in temperature, the Marangoni force is exerted on the surface of molten material, as shown in Fig. 2(h). Thus, according to the ring-shaped intensity form, the Marangoni force acts as a central force to transport the molten material to the central laser axis, and then pushes up to form a protrusion, which cannot be deformed further after solidification. Note that the force can displace the molten metal also towards the positive r direction. In fact, as shown in Figs. 2(d)–(f), there was a transported matter around r = 8 µm. However, it was small, because we employed the ring-shaped heating and the cylindrical coordinates with the axial symmetry. The larger mass transport was required to build up a large structure outside than inside.

Fig. 2. (a)-(f) Typical time-evolution of simulation with the parameters w₀ = 6 µm, γ = −0.3 mN/m·K, μ_l = 1 mPa, and P = 1820 W. (g) Heat source and (h) Marangoni force at each time were also shown. The solid (liquid) region in (a)-(f) is indicated by the gray (red) color according to α_s – α_l. The upper white region was filled with the gas state. The interface between the gas and metal (solid and liquid) is indicated by the black contour of α_g = 0.5. (g)(h) The value of Q_opt and r-component of F_M was integrated over z; therefore, the original volume heat source and volume force can be recognized as an absorbed intensity of the laser and a surface stress. We assumed the surface tension of the solid state to be zero. Also see the Visualization 1.

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Figure 3 shows the height of protrusion at r = 0, that here termed center height, produced by doughnut beam for the different parameters of beam and material: spot radius w₀, viscosity µ, and surface tension coefficient γ as a function of the laser power. The employed combination of condition of µ and γ is summarized and labeled as {1}–{5} in the inset table. We employed a different power range for each value of w₀, based on the light intensity required to melt and drive the material. The center height became the maximum at the specific power. This behavior can be easily understood from the Visualization 2, where the time-evolution of material shape was shown for the condition of w₀ = 7 µm and {1}, and the different powers: P = 2500, 2750, and 3000 W from left in the Visualization 2. The result of P = 2750 W is corresponding to the maximum height of w₀ = 7 µm and {1} in Fig. 3(a). In the low power situation of P = 2500 W, the molten material solidified before reaching to the center, like the burr commonly observed in laser processing. Then, the increase of power enables the molten material to approach the center and the maximum height up to P = 2750 W. Meanwhile, in the higher-power region, the solidification of molten material took time. When such a solidification process was slow, the protrusion was collapsed to be disappeared as shown in the result of P = 3000 W in the Visualization 2. Thus, the power dependence of the center height exhibited a peak profile. To simplify the simulation, some processes such as ablation were not included. Thus, also in the higher power region, the molten material could be possibly cooled to form the protrusion, if the ablation removes the material with large excess energy.

Fig. 3. Power dependence of center height of protrusion h_c as a function of power P at the final time. In both (a) and (b), the spot radius was varied with w₀ = 5, 6, 7, and 8 µm, with the power (kW) range [0.9, 1.4], [1.6, 2.2], [2.4, 3.6], and [3.6, 5.0], respectively. The step size was 20 W for w₀ = 5, 6 µm, and 50 W for 7, 8 µm. As shown in the above table of parameter conditions, the surface tension coefficient and the viscosity of molten metal were fixed or varied. The condition {2} was same in both (a) and (b).

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In addition, the parameters w₀, γ, and µ varied to control the height of the produced protrusion. For example, as the spot radius w₀ gets larger, the doughnut beam drives the larger volume of material and the size of protrusion increases by heating and melting in the wide area. Furthermore, as shown in Fig. 3(b), the Marangoni stress depends on the surface tension coefficient γ of the molten material, and changes the protrusion height. As shown in Fig. 3(a), lower viscosity µ also leads to the formation of a higher protrusion, because the shear stress affecting the motion of the molten material decreased. In either case, the conditions for the higher fabricated protrusion would decrease the laser power for the transport of the molten material toward the center. As described before, if the material reaching the center cannot be solidified, the protrusion disappears, and its height decreases. In Fig. 4, the shape of produced protrusion at the power of maximum height of {1} was plotted at each final time of simulation. Also, the time evolution can be confirmed in Visualization 1. Furthermore, referring the vertical lines of r = 200 and 400 nm in Fig. 4, there are the conditions to obtain the sharp nanostructure.

Fig. 4. Material shape at final time for the parameter condition {1} shown in Fig. 3. The results correspond to the power of peak height for each spot radius w₀ of {1} in Fig. 3(a). The final simulation time was 300, 400, 500, and 600 ns for w₀ = 5, 6, 7, and 8 µm, respectively. The vertical dashed lines indicated r = 200 and 400 nm. Also see the Visualization 1.

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As for the viscosity µ, the fitting of experimental result was reported [38]. If it is valid also in the higher temperature region of liquid phase, µ = 1, 3, and 5 mPa·s of {1} – {3} are given at the temperature 4540, 3800 and 3530 K, respectively. In addition, while the reported value of γ around the melting point was ranged from −0.21 to −0.29 mN/m·K [38–40] as mentioned above, the value of γ = −0.3 mN/m·K of {1} – {3} is close to reported values. Thus, under these realistic conditions, Fig. 4 indicated that such the protrusion structure could be actually formed based on the principle discussed in this study. In addition, if the molten material has the low viscosity like {1} or the larger temperature dependence of surface tension like {5}, the higher structure could be obtained. Particularly, some results such as Figs. 3(a) and (b) indicated the possibility of producing very sharp nano-needle within 200 nm in radial direction. It is surprising that the sharpness of such a structure is beyond the diffraction limit of light even in the case of the loose focused beam of spot radius larger than 5 µm providing the high thermal effect over the wide area.

4. Conclusion

Based on the thermo-fluid simulation assuming the pulsed doughnut beam, we demonstrated the formation process of the nano-needle structure via the Marangoni effect, resulting from the ring-shaped intensity distribution. In comparison with processing using ultrashort laser pulses, nanosecond pulsed LG beam without OAM can induce Marangoni stress on the molten material and generate a useful nanostructure for various applications, which is conventionally considered to be an unnecessary burr. In particular, the obtained results clearly showed the possibility that a high photothermal effect with nanosecond pulsed laser can be utilized for the precise production of sharp nanostructures with a few hundred nanometer-size under carefully tuned conditions, such as the intensity distribution and pulse width of laser, and the viscosity and surface tension of liquid phase of material. Irradiation with a ring-shaped intensity distribution could form a protrusion nanostructure in molten material via surface tension, which leads to the Marangoni effect. The obtained sharp nanostructure would be used for the dramatic sensitivity enhancement in optical condensation biosensors to detect nanoscale biomaterials and microbes such as viruses and bacteria [44,45]. In particular, the principle discussed in this study could be applicable for the various materials such as metal, semiconductor, and biological materials. If the contribution of the optical force with angular momentum of circularly polarized optical vortices can be clarified, nanostructures with arbitrary chirality can be fabricated leveraging designed structured beams. Further understanding of these phenomena will open an avenue for unconventional optical nano-processing for the creation of structured nanomaterials by transferring many degrees of freedom of various structured light beams.

Funding

Japan Society for the Promotion of Science KAKENHI Grant-in Aid for Scientific Research on Innovative Areas (JP16H06507); JSPS KAKENHI Grant-in Aid for Early-Career Scientists (JP20K15196); JST-Mirai Program (JPMJMI18-GA, JPMJMI21G1). Also, the authors are partially supported by JSPS KAKENHI (JP17H00856, JP21H04964, JP17K19070, JP18H03884, JP22H05138, JP21H01785); JST-Core Research for Evolutional Science and Technology (JPMJCR1903); Key Project Grant Program of the Osaka Prefecture University.

Acknowledgments

The authors would like to thank Prof. H. Ishihara and Prof. S. Tokonami for their support and encouragement.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Name	Description
Supplement 1	Parameters used in this study
Visualization 1	Movie corresponding to Fig. 2. and Fig. 4.
Visualization 2	Movie corresponding to Fig. 3

Nanoneedle formation via doughnut beam-induced Marangoni effects

Abstract

1. Introduction

2. Simulation method and model

3. Results and discussion

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (3)

Data availability

Cited By

Figures (4)

Equations (12)

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