The evolution of the shape and size of a bubble around a nanowire immersed in a liquid can be studied as a light absorption problem and consequently can directly be related to the distribution of the temperature around the nanowire. Such a physical phenomenon can be seen as the photo-thermal coupled problem of nanowire illuminated by an electromagnetic wave. The resolution of the multiphysic model allows to compute the variation of the temperature and consequently the evolution of the created bubble. An advanced adaptive remeshing process is developed to solve the numerical model using Finite Element Method. An optimization process is applied to solve the coupled problem and is used to detect the size of the produced bubble around nanowire under illumination. The adaptive remeshing process permits to control the convergence of the numerical solution relatively to the evolution of the temperature field. The process allows to study the evolution of the shape and size of the bubble. We show the influence of the laser parameters on the evolution of the bubble. The informations about the geometry of the nanowire can be deduced from the size and shape of the bubble.
© 2013 OSA
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 . 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, Hz(x, y)) (resp. E(x, y) = (0, 0, Ez(x, y)), must be computed in the domain Ω. The Hz (resp. Ez) component of the magnetic field (resp. electric field) is deduced by solving the scalar equation, respectively:10, 11, 12]. The condition that satisfies these constraints, by illuminating along the y-axis, is written as follows: Eq. (3) (resp. Eq. (4)) and Hi = H0 exp(jk0x)/(cμ0) (resp. Ei = E0 exp(jk0x)). By using the Maxwell-Ampere equation in TM mode, the electric field E(x, y) can be written as follows : Eq. (2) in mode TE. The TMy, TMx, TEy and TEx 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:
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 .
- 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: 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 .
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 Cp(Ω), defined through an a posteriori error estimator related on the interpolation error, is:8, 19]. A new mesh Mp(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 : the first one CQ(Ω) related to the heat source Q and the second one CT(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.
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 (εr(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 εr(TiO2)950 = 5.50007 + j0.00164. We study the influence of the illumination modes, the power density per area units (PS) 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 PS = 1.75 × 1012W/m2, the results of the adaptive process on mesh and temperature maps (with γ = 0.0001, hmax = 40 nm and hmin = 0.03 nm), are presented on Figs. 2. The use of the adaptive process on the temperature field T produce the mesh MF and the temperature map. By fixing the vapor threshold α = 100°C (373.15K), Figs. 2(a) and 2(b) show the mesh MF after bubble detection for the two different illumination modes TEy and TMy, 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 εr(vap) = 1.79 and thermic conductivity κ(vap) = 0.05 Wm−1K−1 are required to compute the temperature on the adapted mesh MF. Figures 2(c) and 2(d) present the distribution of the temperature field T on MF after convergence to a stable solution for two modes TEy and TMy, 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 TEy is larger than in mode TMy. 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 Rn = a/b for modes of illumination: TEy, TMy, TEx and TMx. The temperature increase quasi linearly as function of Rn for the different modes.
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 TEy is more efficient than TMy to obtain a largest bubble and to detection. That TEy 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 TEy polarized laser pulse in water at the temperature T0 = 25°C with three different power densities per area units PS = 1.7 × 1012W/m2, PS = 1.9 × 1012W/m2 and PS = 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 PS values are closed to experimental Nd-YAG laser with 20W-10kW power range and size spot radius of 2 – 100μm. When PS increases, the heat source Q increases, then the temperature and also the size and shape of bubble increase as function of the aspect ratio Rn = a/b. The evolution of the aspect ratio of the bubble Rb = A/B (A and B are the semi-axes of the bubble) as function of the aspect ratio of the nanowire Rn for three different laser power densities is shown on Fig. 4(a). The bubble aspect ratio Rb decreases when the nanowire aspect ratio Rn 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 Rb 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: Vb = πAB) as function of the volume of the nanowire (Vn = πab) for the same power densities. From the computed data, a function f (satisfying Rb = f (Rn)) and a function g (satisfying ln(Vb) = g(ln(Vn))) 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: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(Ps), C = L2(Ps) and (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: Table 2. Therefore, two functions F and G of two variables can be deduced:
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 TEy polarized laser pulse with power Ps = 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 Rn. The evolution of the aspect ratio of the bubble Rb as function of the aspect ratio of the nanowire Rn for three initial temperatures is shown on Fig. 6(a). Figure 6(b) presents the evolution of the bubble volume Vb as function of the volume of the nanowire Vn. 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 (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 ( , ) of the applications L1, L2 and are presented in Table 4. With such parameters, two functions of two variables are obtained:
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(TEy) and T(TEx) for TE mode (resp. between T(TMy) and T(TMx) 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 Q3D/Q2D is 1.19 for R0 = 170 nm and the minimum 1.07 is reached for R0 = 50 nm. Consequently, as indicated in Ref. , T3D/T2D ≈ (1/3) * [Q3D/Q2D] 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%.
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.
The authors thank the ANR-2011-NANO-008 NANOMORPH for financial support.
References and links
1. M. Boroski, A. C. Rodrigues, J. C. Garcia, L. C. Sampaio, J. Nozaki, and N. Hioka, “Combined electrocoagulation and TiO2photoassisted treatment applied to wastewater effluents from pharmaceutical and cosmetic industries original,” J. Hazard. Mater. 162, 448–454 (2009). [CrossRef]
2. D. Karamanis, A. N. Okte, E. Vardoulakis, and T. Vaimakis, “Water vapor adsorption and photocatalytic pollutant degradation with TiO2-sepiolite nanocomposites original,” Appl. Clay Sci. 53, 181–187 (2011). [CrossRef]
3. G. Bystrzejewska-Piotrowska, J. Golimowski, and P. L. Urban, “Nanoparticles: their potential toxicity, waste and environmental management,” Waste Manage. 29, 2587–2595 (2009). [CrossRef]
5. D. Lapotko and E. Lukianova, “Laser-induced micro-bubbles in cells,” Int. J. Heat Mass Transf. 48(1), 227–234 (2005). [CrossRef]
6. D. Lapotko, E. Lukianova, and A. Shnip, “Photothermal responses of individual cells,” J. Biomed. Opt. 10, 014006 (2004). [CrossRef]
7. D. Barchiesi, T. Grosges, E. Kremer, and M. Lamy de la Chapelle, “Electromagnetic heat induced in meso-structures: computation of temperature in metallic dimers,” PIERS Online 7, 406–410 (2011).
9. M. Born and E. Wolf, Principle of Optics (Pergamon, 1993).
10. J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, 1993).
11. T. Grosges, S. Petit, D. Barchiesi, and S. Hudlet, “Numerical modeling of the subwavelength phase-change recording using an apertureless scanning near-field optical microscope,” Opt. Express 12, 5987–5995 (2004). [CrossRef] [PubMed]
12. T. Grosges, A. Vial, and D. Barchiesi, “Models of near field spectroscopic studies: comparison between finite element and finite difference methods,” Opt. Express 13, 8483–8497 (2005). [CrossRef] [PubMed]
13. R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Amer. Math. Soc. 49, 1–23 (1943). [CrossRef]
14. P. Silvester and G. Pelosi, Finite Elements for Wave Electromagnetics: Methods and Techniques (IEEE, 1994).
15. D. Barchiesi, E. Kremer, A. Cherouat, T. Grosges, and H. Borouchaki, “Dilation of nanonatennas induced by an electromagnetic source,” Adv. Electromagn. 1, 48–57 (2012). [CrossRef]
16. I. Stakgold, Boundary Value Problems of Mathematical Physics (Macmillan, 1968), vol. I-II.
17. P. G. Ciarlet, Basic Error Estimates for Elliptic Problems (North Holland, 1991).
18. D. Xue and L. Demkowicz, “Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method,” Finite Elem. Anal. Des. 42, 570–579 (2006). [CrossRef]
20. R. Radovitzky and M. Ortiz, “Error estimation and adaptive meshing in strongly non-linear dynamic problems,” Comput. Meth. Appl. Mech. Eng. 172, 203–240 (1999). [CrossRef]
21. M. Ainsworth and J. T. Oden, “A posteriori error estimation in finite element analysis,” Comput. Meth. Appl. Mech. Eng. 142, 1–88 (1997). [CrossRef]
22. T. L. Brown and J. A. Rice, “The effect of laser wavelength and power density on the laser desorption mass spectrum of fulvic acid,” Org. Geochem. 31, 627–634 (2000). [CrossRef]
23. K. O’Connor, O. Morris, and E. Sokell, “Angular and energy distribution of Sn ion debris ejected from a laser-produced plasma source, for laser power densities in the range suitable for extreme ultraviolet lithography,” J. Appl. Phys. 109, 073301 (2011). [CrossRef]
24. D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441 (1963). [CrossRef]
25. P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numer. Anal. 15, 977–992 (1978). [CrossRef]