## Abstract

The development of methods to measure the size of nanoparticles is a challenging topic of research. The proposed method is based on the metrology of the stable vapor bubble created by thermal coupling between a laser pulse and the nanoparticle in a droplet. The measurement is realized by digital in-line holography. The size of the nanoparticle is deduced from numerical simulations computed with a photo-thermal finite element method.

© 2015 Optical Society of America

## 1. Introduction

Nowadays nanoparticles are used in many products and applications [1–3 ] and therefore the production volumes of nanoparticles are growing. The quality control of these nanoparticles needs instruments and methods which can measure the size of nanoparticles in a short time, with good repeatability and at a low cost. Many studies have investigated the size measurement of nanoparticles [4–9 ]. We admit that much high resolution measurement spends much time consuming and high cost instruments. Here, a new methodology of measurement is presented. Due to the difficulty to measure nanoscale, indirect measurement of the size of nanoparticles is proposed. The method is based on the measurement of the microscale bubble induced by heating the nanoparticles in water using Digital in-line holography (DIH) technique [10, 12]. The reconstruction process from the DIH holograms of bubble by means of the fractional Fourier transformation provides information on size, shape and location of bubbles in a droplet. The size of nanoparticles that produced bubbles is deduced and obtained from numerical simulations of the bubble formation [13, 14]. The physical basis of the bubble formation around the nanoparticles or aggregate of nanoparticles, is the exceeding of the threshold of water vaporization, induced by sufficient absorption of an electromagnetic wave by the particle (laser-induced nucleation). The emergence and the size of the bubble depend on the laser power but also on the material and the size of the nanoparticles or aggregate of nanoparticles. In the experimental process, the aggregate of nanoparticle is considered as a nanoparticle. Due to the aggregation grew size by time, thus our nanoparticle solution is treated with a short time of preparation and under ultrasonic wave machine before using. This technic is studied on a low concentration of nanoparticle solution and for high concentrations, other technics exist [15, 16]. Moreover it is a dynamic process. The paper is organized as follows. The first section presents the principles of DIH. The reconstruction of the bubble size from DIH holograms is presented in the second section. The third section is devoted to the numerical method used for the recovering of the nanoparticle size before concluding.

## 2. Measurement of a vapor bubble as inclusion in a pure water droplet

The experimental setup composed of two sub-parts: a laser-induced nucleation experimental setup and a digital in-line holography experimental setup (Fig. 1).

#### 2.1. The laser-induced nucleation experimental setup

The laser-induced nucleation experimental setup is in charge of producing the bubbles around nanoparticles. The Nd:YAG laser pulse at 532nm with the pulse width *w _{t}* = 5ns and the pulse repetition rate at

*PRR*= 15Hz is used. The beam is focused by the lens of focal length

*f*= 500mm before a pure water droplet (

*H*

_{2}

*O*) seeded with randomly positioned nanoparticles. The beam waist is far from the droplet that contains nanoparticles suspension. It is located at a distance 73.5mm from the droplet to avoid the evaporation of the pure water. The diameter of the Nd:YAG beam, denoted by

*D*, at the center of the droplet is estimated to

_{l}*D*= 1.2mm. The power density per area units, denoted by

_{l}*P*, is defined by

_{s}*P*is the mean power from the output of laser. In our case,

_{m}*P*= 0.125W and then

_{m}*P*is equal to 1.47

_{s}*×*10

^{12}

*W/m*

^{2}. Note that

*P*is an input for simulations of bubble formation. A heating point is created in each nanoparticle or aggregate of nanoparticles by laser illumination. The single-pulsed operation mode of Nd:YAG laser is used. Consequently a surrounding bubble grows with time, up to reach a suitable diameter of several micrometers. Here, the considered measured bubble is a stable bubble reaching equilibrium size. The combination of short laser pulses and relaxation times induce transient bubbles that are created after each shot, they collapse before become a stable bubble. Such transient bubbles expand and collapse in a few microseconds [17, 18]. After several laser pulses the temperature of the nanoparticle reachs an equilibrium and the bubble becomes a stable vapor bubble. For nanoparticle (e.g.

_{s}*T*2) and with the experimental and physical parameter values, the theoretical time for a stable vapor bubble, denoted

_{i}O*τ*and the maximum temperature in the nanoparticle, denoted

_{bubble}*T*are given by:

_{max}*R*is the radius of the bubble,

_{bubble}*T*= 373

_{boil}*K*is the boiling temperature,

*T*is the maximum temperature at the interface between the nanoparticle and the vapor,

_{max}*n*and

_{t}*w*are the number of pulse shots and the pulse width,

_{t}*ρ*(.),

*C*(.),

_{p}*κ*(.) are the mass density, the specific heat and the heat conductivity of materials, respectively. Here, the time to reach the equilibrium radius

*τ*is varying from 0.5 to 16 ms (depending on the absorbed energy and the temperature elevation

_{bubble}*T*in the nanoparticle) and necessitates an illumination duration

_{max}*n*of about 1 second. For a value of

_{t}w_{t}*τ*16

_{bubble}≈*ms*and a measured bubble of radius

*R*= 6

_{bubble}*μm*, the temperature

*T*460

_{max}≈*K*is obtained with about 10 laser pulses of 5ns. The measurements of the bubble size evolution for successive imaging times show no variation of the bubble size which is the characteristic from a stable bubble. For a collection of different particle sizes, (the absorbed energy and the size of the nanoparticle are related with the

*T*in the nanoparticle), the time to reach equilibrium radius of the bubble is varying from 0.1 to 20.0 milliseconds with a few (10 to 100) pulses of 5ns. In this setup with the droplet of the

_{max}*TiO*

_{2}nanoparticles solution in very low concentration (0.05% volume/volume), one or two single pulses are enough for the bubble formation. This magnification allows us to apply the digital in-line holography technics.

#### 2.2. The DIH experimental setup

The DIH experimental setup uses an HeNe laser (wavelength 632.8 nm) to probe the bubbles in the droplet. The beam waist (*ω*
_{0} = 2.5*μ*m) of the laser beam is in the focal plane of a lens of focal length *f*
_{0} = 56mm (*e*
_{0} = *f*
_{0}). Then *e*
_{1} = 242*mm* away, the beam illuminates the droplet of the nanoparticle solution after passing through a first micro-objective of focal length *f*
_{1} = 56mm. This droplet which has a diameter of 2.6 mm along the *x*-axis *y*-axis is static vertically under the tip of a needle and its location is on the cross section of Nd:YAG line and holographic HeNe laser line. The other terminal of this needle is connected to the syringe which contained the nanoparticle solution and the digital pump for ejecting liquid. The distance between the first micro-objective and the droplet is *e*
_{2} = 10.65mm. After interaction with bubbles in the droplet the light pass through a second micro-objective, *f*
_{2}, with the same focal lens *f*
_{1} (*e*
_{3} = 5.75mm), and finally the resulting signal comes on a CCD sensor at a distance *z* = 39.3*mm* from the a second micro-objective. The pixels sizes of the CCD sensor are 4.4*μ*m. The characteristic of the micro-objective, *f*
_{1} is NA=0.25. With *λ* =632.8nm and a width of the input beam (collimated beam) *w* = 15*mm*, from the diffraction limited, *d _{limit}* =

*λ*/2

*NA*. We obtain a resolution close to 632.8

*nm*/(2

*×*0.25)=1.28

*μ*m in the plane just before the droplet. Then the beam width at the input plane of the second micro-objective,

*f*

_{2}, is greater than the aperture of the second micro-objective. This second micro-objective allows us to eliminate the high frequency noise contains in the beam and allows us to select the optical signal close to the optical axis (Fresnel’s approximation). The propagation of light from the laser source to the bubble and then from the bubble to the CCD sensor can be described by two matrices M

*and M*

_{i}*respectively [19–21 ].*

_{t}The optical system being axisymmetric the incident matrix is the same in all directions:

The transmitted matrix is:

*d*is the diameter of the droplet,

_{r}*δ*= 1.9mm is the approximated position of inclusion in the droplet,

*n*and

_{d}*n*are the optical indexes of water and air and the diameter of the droplet is

_{a}*d*= 2.7mm. An example of the CCD signal generated by a vapor bubble created by Nd:YAG pulses is given in Fig. 2. A fringe pattern with high contrast appears clearly in the figure, showing the interference between the reference and the scattered wave. The next step is to get geometrical information on the bubble. To do this, a digital reconstruction of the image of the object is realized by means of a mathematical operator: the fractional Fourier transformation.

#### 2.3. Reconstruction by fractional Fourier transformation

The fractional Fourier transform (FRFT) is used to reconstruct the image of an object from the intensity distribution in the hologram obtained from digital in-line holography [10]. The intensity in holograms is a function of *r*, the distance to its center, due to the axial symmetry of the fringe pattern. The reconstructed signal depends on distance *ρ* to the center of the image. The FRFT of fractional order α ∈ ℂ of an function *f* (*r*) is defined as [22–24
]:

*i*is the pure imaginary unit such as

*i*

^{2}=

*−*1, ${s}^{2}=N\cdot {\delta}_{p}^{2},$

*J*

_{0}is the zero order Bessel function of the first kind and

*a*is a parameter that has to be determined along each direction

*x*, (

*a*) and

_{x}*y*, (

*a*). The best parameter set is obtained for two fractional orders along

_{y}*x*and

*y*directions

*a*and

_{ox}*a*. The reconstruction by means of FRFT consists in canceling quadratic phases contained by the intensity distribution of the hologram and at the same time to realize a classical Fourier transformation. As the intensity is versus the Fourier transform of the object, here the bubble, and as the double Fourier transformation of a function is the function, we can retrieve the image of the object. Then, the reconstructed bubble inclusion image from the experimental diffraction pattern in Fig. 2 is shown in Fig. 3. This optimal reconstruction is realized with the optimal fractional order

_{oy}*a*=

_{ox}*a*= 0.6735. The center of the reconstructed image of the bubble is white and therefore the image is characteristic of the transparent bubble and not from an opaque particle. The spot of light at the center of the reconstructed image predicted by Lorenz Mie theory has been experimentally observed as we can see in the following reference [25]. Depending on the bubble shape, the bubble could be slightly elliptical, then two diameters along the major and minor axis, denoted

_{oy}*D*and

_{est(x)}*D*, can be extracted. In this optical system a magnification scaled, denoted

_{est(y)}*G*, is applied on the estimated diameter such as [11] where

*D*is the real diameter of the bubble and

_{th(i)}*D*, the estimated diameter measured in the reconstructed image (

_{est(i)}*i.e.*in 3). The magnification scaled is defined by

Note that the magnification is the same for the two axis because the optical system is axisymmetric. With Eq. (9) and by means of the parameters of the optical system, the magnification factor *G* is equal to *−*0.257. The profile of the reconstruction shown in Fig. 4, allows us to estimate the diameters along *x*-axis and *y*-axis. With the knowledge of the pixel size (i.e. 4.4*μm*) and the number *N* of the pixels, we can estimated *D _{est}*

_{(}

_{x}_{)}=

*D*

_{est}_{(}

_{y}_{)}= (21 1) pixels (i.e (92.4

*±*4.4)

*μm*). Finally, by means of Eq. (9), the real diameters are estimated

*±*to

*D*

_{th}_{(}

_{x}_{)}=

*D*

_{th}_{(}

_{y}_{)}= (23.7

*±*1.1)

*μm*. Now, we present a droplet with many bubbles. One sequence image of hologram is selected and shown in Fig. 5. In this experiment, the distance

*z*from the second micro-objective to the CCD sensor is 40.43mm. The results of the optimal fractional orders,

*a*, the position,

_{ox},a_{oy}*δ*, the diameter of a droplet,

*d*

_{(}

_{i}_{)}, the estimated diameter,

*D*and the real diameter,

_{est(i)}*D*of each bubble with the tag number,

_{th(i)}*j*, in Fig. 5 are given in Table 1. The bubble dimensions being measured, that of the nanoparticles are still to be deduced. For this simulation results are generated from a physical model of bubble formation around nanoparticles. Furthermore, the finite dimensions of the pixels and the coherence here are not taken into account [26].

## 3. Numerical results and discussion

In this experiments, *TiO*
_{2} spherical nanoparticles of radius *R _{particle}* were investigated. The complex relative permittivities at wavelength

*λ*= 532 nm are

*ε*(

_{r}*TiO*

_{2})

_{532}= 6.1000+

*i*0.00395 and

*ε*(

_{r}*water*) = 1.79 The thermal conductivities are

*k*(

*TiO*

_{2}) = 11.7

*Wm*

^{−1}

*K*

^{−1}and

*k*(

*water*) = 0.6

*Wm*

^{−1}

*K*

^{−1}at temperature

*T*

_{0}= 25°

*C*(298.15

*K*). All materials are considered isotropic and homogeneous. Finite element model is used to compute the time evolution of the bubble radius as a function of the laser power and of the radius of nanoparticle. The 3D model and the numerical method is fully described in [13, 14]. Figure 6 shows the relationship between the radius of the bubble and the radius of the particle for three illumination power

*P*. The logarithm of volumes follows a linear rule and therefore a simple fitting of curves can lead to the

_{W}*F*-function: ln(

*V*) =

_{bubble}*F*(ln(

*V*)). This function has mathematical but not physical sense even if it results from physical model. Consequently the data are considered as dimensionless to find simulation values and behavior laws. This approach is well known in engineering of complex systems. The

_{particle}*F*-function is continuous and strictly increasing, therefore the inverse function

*F*

^{−1}also exists. Therefore the measurement of the bubble volume

*V*can be used to determine the volume of the nanoparticle

_{bubble}*V*through the relation ln(

_{particle}*V*) =

_{particle}*F*

^{−1}(ln(

*V*)). With this function

_{bubble}*F*

^{−1}the radius of the nanoparticle can be related to the radius of the bubble:

*A*and

*B*are the fitting parameters. From this function, a mathematical relationship between the radius of the nanoparticle and the radius of bubbles can be deduced:

Moreover, the parameters *A* and *B* can also be expressed as linear functions of the laser power *P _{W}* :

*A*= (

*a*

_{1}

*P*+

_{W}*b*

_{1}) and

*B*= (

*a*

_{2}

*P*+

_{W}*b*

_{2}). The fit parameters,

*A*,

*B*,

*a*

_{1},

*b*

_{1},

*a*

_{2}and

*b*

_{2}are given in Table 2. Therefore the measurement of the radius of the bubble

*R*can be used to calculate the radius

_{bubble}*R*of the spherical nanoparticle:

_{particle}*R*= (11.85

_{bubble}*±*0.55)

*μm*then

*R*is find to be (27.45

_{particle}*±*0.50)

*nm*(i.e.

*D*= (54.9

_{particle}*±*1.0)

*nm*). This result can be compared to the knowledge on the nanoparticles in water: their mean radius is 50nm for this test sample. Therefore we can conclude that the method can be considered as efficient to obtain the size of nanoparticles by the indirect measurement of that of surrounding bubbles. In the same way, the hologram of many bubbles which is presented in Fig. 5 is analysed with the same process as illustrated in Fig. 2, Fig. 3 and Fig. 4 and the estimation of the size of the nanoparticles is possible with Table 2. Then the estimation of the diameter of the nanoparticles is given in Table 3.

## 4. Conclusion

The paper focuses on the recovering of the sizes of spherical *TiO*
_{2} nanoparticles in low concentration from the sizes of the bubbles created by photothermic process. The metrology of the vapor bubble is achieved from an in-line digital holography and the size of the bubble is recovered. By solving an inverse problem, the size of the nanoparticles can be related to the size of the produced bubbles. The influence of the laser power related to the sizes of the bubble and the nanoparticle is also presented. The advantage of the method and its ability to take into account the shape (ellipticity) of the bubble, would permit to extend the domain of application to the computation of non spherical nanowires.

## Acknowledgments

The authors thank the French National Agency under grant ANR-2011-NANO-008 NANOMORPH for financial support.

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