Nanobubble evolution around nanowire in liquid

Anis Chaari; Thomas Grosges; Laurence Giraud-Moreau; Dominique Barchiesi

doi:10.1364/OE.21.026942

1. Introduction

Over the past decades, a growth of research activities has been achieved in nanoscience and industrial sectors (in optic, thermic, electronic and mechanic). Titanium dioxide (TiO₂) and Zinc oxide (ZnO) have been the most used nanomaterials in the chemical industry and in the manufactures of nanotubes and nanowires since their commercial production in the early twentieth century [1, 2]. Then, the studies of these properties are necessary in order to evaluate their effects on the environment [3, 4]. The detection and the analyze of the presence of toxic substances are crucial. Only few measure instruments enable to detect and identify particles at nanometer scale. Moreover, many of such instruments are expensive in term of costs and their uses can be difficult due to a weak signal to noise ratio. Therefore, the methods of detection can be very difficult to apply in industrial context. Here, an indirect method is investigated and consists in studying the presence of a nanowire in a liquid by detecting the formed bubble. This method is based on the analysis of the thermic response of nanowires immersed in water under illumination and related to the bubble produced. In fact, the nanoparticle, illuminated by an electromagnetic wave, absorbs the energy and consequently it heats. When the temperature overpass the threshold of vaporization of water, a nanometric bubble is formed around the nanowire. When an increase of the mean temperature of the nanomaterial occurs, the size of nanobubble grows before being detected [5, 6]. The identification of the size and shape of the created bubble allows, after solving the inverse problem, to analyze the morphology and the properties of the nanowire in order to deduce informations about nanowire toxicity. The numerical model associated to the problem is reduced to solve a light-mater-thermal coupled system as function of the geometric parameters of nanowires (size and shape) and physical parameters (i.e. laser wavelength, laser power, permittivity of materials, material conductivity). The photo-thermal response of the nanowires illuminated by an incident laser radiation is studied through a multiphysics model solved numerically. The behavior of the bubble formed around a nanowire of TiO₂ in water and illuminated by a laser field is analyzed. An adaptive remeshing process coupled with an optimization loop enables to capture the evolution of the size and the shape of the bubble. The process ensures the convergence of the approximate numerical solution to the physical solution [7, 8].

The structure of that paper is the following: The formulation of equations of numerical problems and the description of the resolution method through the remeshing process, are developed in Sec. 2. The analysis of simulation results and the discussion about of the performance of the method is achieved in Sec. 3 before concluding in Sec. 4.

2. Formulation of the problem, method of resolution and adaptive remeshing process

In that section, we present the differential equations associated to light-matter-thermal coupled problems, the numerical method of resolution and the adaptive remeshing loops.

2.1. Electromagnetic and thermal equations

Maxwell’s equations are a set of fundamental equations governing electromagnetic phenomena. These equations can be written in both differential and integral form and are solved through numerical methods. The used numerical method is based on the resolution of differential equations. Assuming a time harmonic dependence of the form exp(jωt) for the fields, where ω is the angular frequency of the harmonic wave, the partial differential equations to be solved are the Helmholtz equations for the magnetic H and electric E fields [9]. Here, we study the particular case of 2D structures (nanowires, nanotubes...) along the z-axis. Two modes of polarized illumination, (transverse magnetic TM, transverse electric TE), are investigated in the resolution of the systems.

In mode TM (resp. TE), the field H (resp. E) is perpendicular to the plane (x, y). Therefore, the problem is scalar and one component of the field, H(x, y) = (0, 0, H_z(x, y)) (resp. E(x, y) = (0, 0, E_z(x, y)), must be computed in the domain Ω. The H_z (resp. E_z) component of the magnetic field (resp. electric field) is deduced by solving the scalar equation, respectively:

[\nabla_{.} (\frac{1}{ε_{r}} \nabla) + k_{0}^{2}] H_{z} (x, y) = 0, \forall (x, y) \in Ω, (TM)

Δ E_{z} (x, y) + ε_{r} k_{0}^{2} E_{z} (x, y) = 0, \forall (x, y) \in Ω, (TE)

where (∇_.) is the divergence operator, Δ the Laplace operator, k₀ = ω/c the wave number of the monochromatic incoming wave, with c the speed of light in vacuum, ω the angular frequency and ε_r the relative complex permittivity of the considered materials which are functions of the spatial coordinates (x, y). To assure the continuity of the normal component of electromagnetic excitations between materials and the uniqueness of the solution H_z(x, y) (resp. E_z(x, y)), a condition on the contour Γ of the computational domain Ω must be imposed [10, 11, 12]. The condition that satisfies these constraints, by illuminating along the y-axis, is written as follows:

\frac{1}{ε_{r}} \frac{\partial H_{z} (x, y)}{\partial n} = - j k_{0} [H_{z} (x, y) - (n_{y} - 1) H_{i} (x, y)], \forall (x, y) \in Γ, (TM)

\frac{\partial E_{z} (x, y)}{\partial n} = - j k_{0} ε_{r} [E_{z} (x, y) - (n_{y} - 1) E_{i} (x, y)], \forall (x, y) \in Γ, (TE)

where ∂/∂n the normal derivative operator, j is the complex number, n_y the normal vector component along the y-axis, H_i = H₀ exp(jk₀y)/(cμ₀) and E_i = E₀ exp(jk₀y) are the incident illumination fields along the y-axis with μ₀ the permeability of vacuum. The illumination along x-axis in TM (resp. TE) mode consists in changing n_y to n_x in Eq. (3) (resp. Eq. (4)) and H_i = H₀ exp(jk₀x)/(cμ₀) (resp. E_i = E₀ exp(jk₀x)). By using the Maxwell-Ampere equation in TM mode, the electric field E(x, y) can be written as follows [10]:

E (x, y) = \frac{- j}{ω ε_{r} ε_{0}} [\nabla \times H (x, y)], \forall (x, y) \in Ω,

where (∇×.) is the rotational operator, ε₀ the permittivity of vacuum. The electric field E(x, y) is deduced directly from the Eq. (2) in mode TE. The TM_y, TM_x, TE_y and TE_x denote modes TM and TE for an illumination along y-axis and x-axis, respectively. The illumination of nanowires by an electromagnetic wave (laser), for the two modes TM and TE, induces three coupled phenomena: the conversion of light into heat, a heat transfer and an increase of the temperature in the nanowire. The source of heat is given by:

Q (x, y) = \frac{ω}{2} ε_{0} Im (ε_{r}) {| E (x, y) |}^{2}, \forall (x, y) \in Ω .

To describe the phenomenon of the temperature distribution, a numerical model for the distribution of the source computed in the nanowire is presented in the form of the equation of heat. Such a parabolic differential equation that describes the temperature variation around the nanowire is written as follows:

[\nabla_{.} (κ (x, y) \nabla)] T (x, y) = Q (x, y), \forall (x, y) \in Ω,

with a boundary condition T = T₀ and κ(x, y) is the thermal conductivity of the materials. In a stationary homogeneous isotropic medium, the solution of the heat equation only depends on the physical parameters of the material (imaginary part of ε_r(x, y) and κ(x, y)) as well as the intensity of the electric field |E(x, y)|².

The resolution of the coupled system: light, nanowire and heat, consists in solving the electromagnetic equation (Eq. (1) or Eq. (2) as function of the mode) and thermic equation (Eq. (7)). The distribution of the temperature is obtained in the computational domain Ω. The size and shape of the bubble is deduced from the temperature map. The spatial area, for which the threshold of vaporization of water α is exceeded, is constitutive of the form of the bubble. A relation between the geometric informations on the bubble and the size of nanowire can be constructed.

2.2. The Finite element method

The numerical method used to solve the coupled system in a domain whose geometry can be complex is the Finite Element Method (FEM). That is used to compute the approximate solutions of the problems with boundary conditions on domains. It has been developed and applied to problem of structural analysis and to problems in the domains of electromagnetic, mechanic and thermodynamic [13, 14]. The resolution of partial differential equation through the FEM is based on the following steps:

The discretization or subdivision of the computational domain Ω, as the the union of sub-domains, constitute the mesh of the domain [10].
The selection of the interpolation functions that provides an approximation of the unknown solution in function of a fixed number of unknown coefficients computed at the nodes of the mesh.
The weak variational formulation of Eq. (1), Eq. (2) and Eq. (7), by using Ritz’s method to compute the magnetic, electric and thermic fields, satisfy the integral equations, respectively: $\int_{Ω} [\nabla_{.} (\frac{1}{ε_{r}} \nabla H_{z} (x, y)) + \frac{ω^{2}}{c^{2}} H_{z} (x, y)] . ν_{1} d Ω = 0,$ $\int_{Ω} [Δ E_{z} (x, y) + ε_{r} \frac{ω^{2}}{c^{2}} E_{z} (x, y)] . ν_{2} d Ω = 0,$ $\int_{Ω} [\nabla_{.} (κ (x, y) \nabla) T (x, y) - Q (x, y)] . ν_{3} d Ω = 0,$ where ν₁, ν₂, and ν₃ are test functions defined on L²(Ω) (the linear space of the scalar functions ν, being 2-integrable on Ω). That formulation provides a finite basis of polynomial functions which presents an approximation of the solutions H_z, E_z and T at each node [15].
The resolution of the system of differential equation with boundary conditions is achieved by using a linear combination of the basic polynomial functions [14, 16]. A linear system is obtained and can be solved by classical numerical methods such as Gaussian, Cholesky or by iterative procedures such as conjugate gradient. The solutions of the problem are computed on each node of the mesh. Moreover, the weak formulation improves the stability of the FEM. The control of the error on the solution decreases the number of nodes and enhances the efficiency of the method for both time and memory resources [10].

2.3. Adaptive remeshing and optimization process

By using the FEM, magnetic, electric and thermal fields are computed on the nodes of the mesh of the computational domain. The systematic increase of the number of nodes conformally to the geometry does not decrease the error between the numerical solution and the physical solution. Therefore, the accuracy of the computed solution depends on the quality of the mesh [8, 17, 18]. Here, we adapt the size of the mesh elements to the physical solution through an adaptive remeshing process [8, 19]. In fact, the objective is to decrease the maximum deviation between the exact solution and the solution associated with the mesh through the interpolation error. That error is based on an estimation of the discrete Hessian of the solution [20, 21]. A physical size map C_p(Ω), defined through an a posteriori error estimator related on the interpolation error, is:

C_{p} (Ω) = {h_{p} (x, y)}, \forall (x, y) \in Ω,

where h_p(x, y) is the physical size defined at each node and is proportional to the inverse of the deviation of the Hessian. For a given maximum tolerance on the physical error γ, the size h_p(x, y) is given by:

h_{\min} \leq h_{p} (x, y) = \frac{γ}{η (x, y)} \leq h_{\max}

where η(x, y) is an estimation of the maximum deviation obtained from the Hessian of the solution and h_min and h_max are the minimum and maximum sizes of the elements. The adaptive remeshing of the domain Ω is generated by the BL2D-V2 software which uses an isotropic or anisotropic meshes of the Delaunay type [8, 19]. A new mesh M_p(Ω) is obtained by remeshing the domain. The approximate solution of the electric field E, the magnetic field H, the heat source Q and the temperature T are computed at each step of the adaption procedure on a mesh through two physical size maps [19]: the first one C_Q(Ω) related to the heat source Q and the second one C_T(Ω) related to the temperature T. The stability of these solutions is ensured even in the zones where a strong variation of fields occurs. The optimization loop with adaptive remeshing process is illustrated in Fig. 1.

Fig. 1 Optimization process and remeshing loops.

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3. Numerical results and discussion

We consider a TiO₂ elliptical nanowire of semi-axes a and b, with thermal conductivity κ(TiO₂) = 11.7 Wm⁻¹K⁻¹ immersed in water (ε_r(water) = 1.79, κ(water) = 0.6 Wm⁻¹K⁻¹). The nanowire is illuminated by a laser, at wavelength λ = 950 nm, with complex permittivity ε_r(TiO₂)₉₅₀ = 5.50007 + j0.00164. We study the influence of the illumination modes, the power density per area units (P_S) and the initial temperature of the water T₀ on the shape and size of the bubble created around the nanowire. The materials of the system are considered isotropic and homogeneous.

3.1. Influence of the illumination modes

For T₀ = 25°C (298.15K), (a = 30 nm, b = 10 nm) and P_S = 1.75 × 10¹²W/m², the results of the adaptive process on mesh and temperature maps (with γ = 0.0001, h_max = 40 nm and h_min = 0.03 nm), are presented on Figs. 2. The use of the adaptive process on the temperature field T produce the mesh M_F and the temperature map. By fixing the vapor threshold α = 100°C (373.15K), Figs. 2(a) and 2(b) show the mesh M_F after bubble detection for the two different illumination modes TE_y and TM_y, respectively. The remeshing process takes into account the shape and size of the bubble. The three materials are presented on Figs. 2(a) and 2(b): TiO₂(red), vapor (green) and water (blue). The mesh is refined on the outline of the nanowire and in the bubble where strong variations of the temperature occur and relaxed inside the nanowire and outside the bubble where the temperature is almost constant. The physical parameters of the vapor such as the permittivity ε_r(vap) = 1.79 and thermic conductivity κ(vap) = 0.05 Wm⁻¹K⁻¹ are required to compute the temperature on the adapted mesh M_F. Figures 2(c) and 2(d) present the distribution of the temperature field T on M_F after convergence to a stable solution for two modes TE_y and TM_y, respectively. In the vicinity of the boundaries of the nanowire and of the bubble (where a strong variation of the temperature is shown), the level curves are well rounded in all domains. Figures 2 show the evolution of the shape and size of the bubble as function of the modes. Due to the mean temperature produced in the nanowire for each modes, the size of the bubble created in mode TE_y is larger than in mode TM_y. The created bubble follows the shape of the nanowire (elliptical) at the beginning and becomes circular with the increase of the temperature. Figure 3 shows the evolution of the mean temperature T in the nanowire as function of the aspect ratio R_n = a/b for modes of illumination: TE_y, TM_y, TE_x and TM_x. The temperature increase quasi linearly as function of R_n for the different modes.

Fig. 2 Adapted meshes M_F (a, b) and temperature maps T (c, d) for TE_y and TM_y mode, respectively.

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Fig. 3 Evolution of the mean temperature as function of the nanowire aspect ratio R_n after creation of the bubble for the four illumination modes.

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We can remark that, for fixed aspect ratio, the mean temperature level in the nanowire is larger with illumination along the y-axis than x-axis. Moreover, that temperature after illumination is also larger in mode TE than in TM. Therefore, the selected illumination mode (TE or TM) and the axis of the illumination (x or y) have an influence on the mean temperature in the nanowire and thus on the shape and on the size of the bubble formed. Here, the use of mode TE_y is more efficient than TM_y to obtain a largest bubble and to detection. That TE_y mode is used in the following:

3.2. Influence of the power density

In order to study the evolution of the shape and size of the bubble as function of the laser power, we consider the nanowire illuminated by a TE_y polarized laser pulse in water at the temperature T₀ = 25°C with three different power densities per area units P_S = 1.7 × 10¹²W/m², P_S = 1.9 × 10¹²W/m² and P_S = 2.1 × 10¹²W/m² [22, 23]. These laser power values are investigative and are choosen in order to be sufficient to initiate a bubble. Nevertheless, these P_S values are closed to experimental Nd-YAG laser with 20W-10kW power range and size spot radius of 2 – 100μm. When P_S increases, the heat source Q increases, then the temperature and also the size and shape of bubble increase as function of the aspect ratio R_n = a/b. The evolution of the aspect ratio of the bubble R_b = A/B (A and B are the semi-axes of the bubble) as function of the aspect ratio of the nanowire R_n for three different laser power densities is shown on Fig. 4(a). The bubble aspect ratio R_b decreases when the nanowire aspect ratio R_n increases. With an increasing in the size of the nanowire, the absorbed energy increases and the maximum temperature also increases (see Fig. 3). The diffusion of the temperature field in the liquid is quasi isotropic and is mainly related to the largest semi-axis of the nanowire. The bubble is only forming in zone where the temperature overpass the liquid vaporization threshold. Therefore, a competition between the decreasing width of the temperature field and the vaporization temperature threshold of the liquid occurs. For a small nanowire, the maximum initial temperature is smaller than for a stretched nanowire and the bubble aspect ratio R_b has not converged to one (i.e. corresponding to a circular bubble). Figure 4(b) shows the evolution of the volume of the bubble (in 2D: V_b = πAB) as function of the volume of the nanowire (V_n = πab) for the same power densities. From the computed data, a function f (satisfying R_b = f (R_n)) and a function g (satisfying ln(V_b) = g(ln(V_n))) can be obtained with a nonlinear least-squares fit method (LLS) by using the Marquardt-Levenberg algorithm [24, 25]. The method is used to find the set of best parameters fitting the data. It is based on the sum of the squared differences or residuals (SSR) between the input data and the function evaluated at the data values. The applied algorithm consists in minimizing the residual variance σ̂₂ = SSR/NDF with NDF the number of degrees of freedom after a finite number of iterations. These functions can be written as follows:

f (x) = A + \frac{B}{{(x - C)}^{2}} and g (x) = A^{*} - \frac{B^{*}}{x - C^{*}},

with (A, B, C) and (A^*, B^*, C^*) the sets of the parameters varying as function of the power densities of the laser. For the function f, the parameter A is related to the maximum ratio of a circular bubble which presents the asymptotical value, B is the inverse of the decay rate of the function which is related to the speed tending to the circular shape and C is the initial ratio from which the bubble begins to form (see Fig. 4(a)). For the function g, the parameter A^* concern the asymptote value which is related to maximum of the logarithm volume of the bubble, B^* is the growth rate of the function which is related to the speed tending to the maximum of the bubble volume and C^* is the logarithm volume of the bubble when the bubble begins to occur. The fit parameters for each power densities of f and g are presented in Table 1. The parameters of fit A, and (A^*, B^*) in Table 1 are almost constant for the values of the power density. We also analyze the other parameters of the functions f and g as function of the laser power, with the fit algorithm. From the computed data (for each fit parameters as function of the power), the parameters can be expressed as a linear function of the power density: B = L₁(P_s), C = L₂(P_s) and

C^{*} = L_{2}^{*} (P_{s})

(see Figs. 5(a) and 5(b)). Three linear relations enable to give the relation between the parameters and the power density and can be written as:

L_{1} (x) = a_{1} x + b_{1}, L_{2} (x) = a_{2} x + b_{2} and L_{2}^{*} (x) = a_{2}^{*} x + b_{2}^{*}

with (a₁, b₁), (a₂, b₂) and (

a_{2}^{*}

,

b_{2}^{*}

) are set of the constant parameters given in Table 2. Therefore, two functions F and G of two variables can be deduced:

R_{b} = [A + \frac{a_{1} P_{s} + b_{1}}{{(R_{n} - a_{2} P_{s} - b_{2})}^{2}}] = F (R_{n}, P_{s})

and

ln (V_{b}) = [A^{*} - \frac{B^{*}}{ln (V_{n}) - a_{2}^{*} P_{s} - b_{2}^{*}}] = G (ln (V_{n}), P_{s}) .

These functions shows that the size of the bubble created around the nanowire depends on the size of the nanowire and on the power density. The f (resp. g) function is continuous and strictly decreasing (resp. increasing), then the inverse function f⁻¹ and (resp. g⁻¹) exists. By using the inverse functions the size and shape of the nanowire can be obtained from the information on the bubble:

ln (V_{n}) = ln (π b a) = ln (π b^{2} R_{n}) = ln (π b^{2} \tilde{F} (R_{b}, P_{s})) = \tilde{G} (ln (V_{b}), P_{s}),

with

\tilde{F} (R_{b}, P_{s}) = {F^{- 1} (R_{n}, P_{s}) |}_{P_{s, fixed}} = [{(\frac{a_{1} P_{s} + b_{1}}{R_{b} - A})}^{1 / 2} + a_{2} P_{s} + b_{2}] = R_{n}

and

\tilde{G} (ln (V_{b}), P_{s}) = {G^{- 1} (ln (V_{n}), P_{s}) |}_{P_{s, fixed}} = [\frac{B^{*}}{A^{*} - ln (V_{b})} + a_{2}^{*} P_{s} + b_{2}^{*}] = ln (V_{n}) .

Consequently,

b = {[\frac{exp (\tilde{G} (ln (V_{b}), P_{s}))}{π \tilde{F} (R_{b}, P_{s})}]}^{1 / 2} and a = {[\frac{\tilde{F} (R_{b}, P_{s})) \exp (\tilde{G} (ln (V_{b}), P_{s}))}{π}]}^{1 / 2} .

Therefore, the measurement of the size and shape of the bubble (R_b and V_b) can be used to define approximately the size and shape of the nanowire (a and b) through the functions F̃ and G̃.

Fig. 4 Evolution of (a) the size ratio of the bubble R_b as function of the aspect ratio of the nanowire R_n for three laser power density of laser and (b) the volume of the bubble V_b as function of the volume of the nanowire V_n.

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Table 1. Fit parameters of functions f and g and associated residual variance σ̂².

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Fig. 5 Evolution of the fit parameters (a) B and C, (b) C^* as function of the power laser density P_s.

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Table 2. Fit parameters of the functions L₁, L₂ and $L_{2}^{*}$ and associated residual variance σ̂².

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3.3. Influence of the initial temperature

Here, we study the influence of the temperature T₀ on the shape and size of the bubble. For that, we consider a nanowire of TiO₂ immersed in water with initial temperatures T₀ = 30°C, T₀ = 40°C and T₀ = 50°C, illuminated by a TE_y polarized laser pulse with power P_s = 1.75 × 10¹²W/m². When T₀ increases, the mean temperature in the nanowire increases, then the size and shape of bubble increase as function of the aspect ratio R_n. The evolution of the aspect ratio of the bubble R_b as function of the aspect ratio of the nanowire R_n for three initial temperatures is shown on Fig. 6(a). Figure 6(b) presents the evolution of the bubble volume V_b as function of the volume of the nanowire V_n. From the computed data (size and volume of bubble for each nanowire and for each initial temperature) and by using the same method and the same algorithm, the same form of functions f and g (Eq. (13)), can be obtained through a fit. The set of fit parameters (A, B, C) and (A^*, B^*, C^*) for the two functions, as function of T₀, are presented in Table 3. The parameters of fit A and (A^*, B^*) of Table 3 are also almost constant for the different values of the initial temperature of water. The use of the fit algorithm through the computed data (for each fit parameters as function of the temperature), allows to find these linear applications B = L₁(T₀), C = L₂(T₀) and $C^{*} = L_{2}^{*} (T_{0})$ (see Figs. 7(a) and 7(b)). The applications enable to define the relation between the parameters and the initial temperature of water. The fit parameters (a₁, b₁), (a₂, b₂) and ( $a_{2}^{*}$ , $b_{2}^{*}$ ) of the applications L₁, L₂ and $L_{2}^{*}$ are presented in Table 4. With such parameters, two functions of two variables are obtained:

R_{b} = A + \frac{a_{1} T_{0} + b_{1}}{{(R_{n} - a_{2} T_{0} - b_{2})}^{2}} = F^{*} (R_{n}, T_{0})

and

ln (V_{b}) = A^{*} - \frac{B^{*}}{ln (V_{n}) - a_{2}^{*} T_{0} - b_{2}^{*}} = G^{*} (ln (V_{n}), T_{0}) .

The size of the nanowire and the initial temperature of the water have an influence on the size of the bubble produced. By using of the same approach developed in Sec 3.2 we obtain:

b = {[\frac{exp ({\tilde{G}}^{*} ((ln (V_{b}), T_{0}))}{π {\tilde{F}}^{*} (R_{b}, T_{0})}]}^{1 / 2} and a = {[\frac{{\tilde{F}}^{*} (R_{b}, T_{0})) \exp ({\tilde{G}}^{*} (ln (V_{b}), T_{0}))}{π}]}^{1 / 2},

with

{\tilde{F}}^{*} (R_{b}, T_{0}) = [{(\frac{a_{1} T_{0} + b_{1}}{R_{b} - A})}^{1 / 2} + a_{2} T_{0} + b_{2}] = R_{n}

{\tilde{G}}^{*} (ln (V_{b}), T_{0}) = [\frac{B^{*}}{A^{*} - ln (V_{b})} + a_{2}^{*} T_{0} + b_{2}^{*}] = ln (V_{n}) .

Consequently, the functions F̃^* and G̃^* allow to determine the size and shape of the nanowire by measuring the size of the formed bubble.

Fig. 6 Evolution of (a) the size ratio of the bubble R_b as function of the aspect ratio of the nanowire R_n for three different inial temperature and (b) the volume of the bubble V_b as function of the volume of the nanowire V_n.

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Fig. 7 Evolution of the fit parameters (a) B and C, (b) C^* as function of the initial temperature T₀.

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Table 3. Fit parameters of functions f and g and associated residual variance σ̂².

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Table 4. Fit parameters of function L₁, L₂ and $L_{2}^{*}$ and associated residual variance σ̂².

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In a practical implementation, the incident illumination, relatively to the orientation of a nanowire, is not perfectly aligned along the x or y direction or with TE or TM polarization. In fact, for a given fixed laser power as shown in Fig. 3, the temperature evolution as function of the incident illumination is a mixing of T(TE_y) and T(TE_x) for TE mode (resp. between T(TM_y) and T(TM_x) for TM mode). All these modes produce similar results that results shown in Figs. 4 and Figs. 6 and can be interpolated by similar F and G functions. The main difference would be a shift in the C and C^* values which are related to the initial ratio from which the bubble is forming (which is related to the minimum laser power inducing bubble). Moreover the presented 2D approach is valid for a problem that is 3D in practice. Indeed, the maximum of the field intensity on the surface of a sphere and of an infinite cylinder of radius R₀ are respectively 2.3 and 1.9 (under TM illumination). The maximum ratio of sources of heat Q₃_D/Q₂_D is 1.19 for R₀ = 170 nm and the minimum 1.07 is reached for R₀ = 50 nm. Consequently, as indicated in Ref. [15], T₃_D/T₂_D ≈ (1/3) * [Q₃_D/Q₂_D] in the interval [0.6;0.77]. Therefore, the results in 2D are of the same order of magnitude as these in 3D problems. The limitation of the size of the nanowire in the third dimension decreases the temperature of about 33%.

4. Conclusion

In this paper we study the evolution of the shape and size of the bubble by solving coupled system light-matter and heat between a nanowire of TiO₂ immersed in water and illuminated by a electromagnetic wave. We show that the size of the produced bubble mainly depends on the mean temperature in the nanowire which is proportional to the power density. An optimization process, coupled with an adaptive remeshing process, is developed to compute the temperature and to determine the evolution of the size of bubble. The adaptive loop process enables to minimize the number of nodes in the computational domain by ensuring the convergence to the stable solution. The influences of the illumination mode, the power density of the laser, the initial temperature of the water and the size of nanowire are analyzed relatively to the size of the bubble. Two functions allowing to relate the size and shape of the bubble to the size and shape of the nanowire are defined. Therefore, the measurement of the size of the nanowire can be reduced to the measurement of the size of the bubble.

Acknowledgments

The authors thank the ANR-2011-NANO-008 NANOMORPH for financial support.

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P_S(W/m²)	Set of parameters of f			${\hat{σ}}_{f}^{2}$	Set of parameters of g			${\hat{σ}}_{g}^{2}$
P_S(W/m²)	A	B	C	${\hat{σ}}_{f}^{2}$	A^*	B^*	C^*	${\hat{σ}}_{g}^{2}$
1.7 × 10¹²	1.00570	0.14968	2.15443	5.60e-05	21.17410	18.05100	5.44644	6.99e-04
1.9 × 10¹²	1.00251	0.11831	1.87168	6.80e-05	21.31800	18.09340	5.30768	1.53e-03
2.1 × 10¹²	1.00397	0.08464	1.65595	5.09e-05	21.29050	18.02770	5.21495	1.92e-03

Parameters of functions	Set of parameters		σ̂₂
Parameters of functions	a	b	σ̂₂
L₁	−0.16530	0.43126	1.60e-06
L₂	−1.23588	4.23564	3.71e-05
$L_{2}^{*}$	−0.57105	6.40982	1.34e-04

T₀(°C)	Set of parameters of f			${\hat{σ}}_{f}^{2}$	Set of parameters of g			${\hat{σ}}_{g}^{2}$
T₀(°C)	A	B	C	${\hat{σ}}_{f}^{2}$	A^*	B^*	C^*	${\hat{σ}}_{g}^{2}$
30	1.00467	0.11874	1.88810	7.07e-05	21.24270	17.49110	5.33605	1.44e-03
40	1.00399	0.07595	1.57577	6.54e-05	21.28880	17.42950	5.17428	2.25e-03
50	1.00074	0.04488	1.22998	2.19e-05	21.11470	17.44720	5.02575	3.86e-03

Parameters of functions	Set of parameters		σ̂₂
Parameters of functions	a₁	a₂	σ̂₂
L₁	−0.00378	0.23097	1.55e-05
L₂	−0.03327	2.89820	8.84e-05
$L_{2}^{*}$	−0.01535	5.79252	1.68e-05

P_S(W/m²)	Set of parameters of f			${\hat{σ}}_{f}^{2}$	Set of parameters of g			${\hat{σ}}_{g}^{2}$
P_S(W/m²)	A	B	C	${\hat{σ}}_{f}^{2}$	A^*	B^*	C^*	${\hat{σ}}_{g}^{2}$
1.7 × 10¹²	1.00570	0.14968	2.15443	5.60e-05	21.17410	18.05100	5.44644	6.99e-04
1.9 × 10¹²	1.00251	0.11831	1.87168	6.80e-05	21.31800	18.09340	5.30768	1.53e-03
2.1 × 10¹²	1.00397	0.08464	1.65595	5.09e-05	21.29050	18.02770	5.21495	1.92e-03

Nanobubble evolution around nanowire in liquid

Abstract

1. Introduction

2. Formulation of the problem, method of resolution and adaptive remeshing process

2.1. Electromagnetic and thermal equations

2.2. The Finite element method

2.3. Adaptive remeshing and optimization process

3. Numerical results and discussion

3.1. Influence of the illumination modes

3.2. Influence of the power density

3.3. Influence of the initial temperature

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (4)

Equations (25)

Optics Express