Temperature Variations of Gold Nanoparticle and Dynamics of Plasmonic Bubble in Water Under Nanosecond Pulsed Laser


Suspended gold nanoparticle in water medium starts to warm up under nanosecond laser irradiation and creates a bubble around itself. The present study aims at evaluating the amount of nanoparticle size reduction at boiling temperature, the temperature variations of the nanoparticle, and its medium and finally the bubble formation moment. To this aim, Mie theory was used to calculate the absorption cross section of the nanoparticle in proximity of the bubble. Heat transfer equations were applied to determine the temperature of the nanoparticle and water. In addition, hydrodynamic equations were initiated to evaluate the expansion of the bubble. Then, these three groups of equations were coupled together and solved numerically. Based on the results, the bubble forms at the critical pressure and consequently due to the slow bubble velocity, temperature gradient in the medium is observed. Further, slight pulse width variations play a significant role on the nanoparticle temperature. The calculation of the nanoparticle heating associated with the creation of the bubble helps in controlling nanoparticle size and understanding the nanoscale heat transfer processes.

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Correspondence to Hamid Nadjari.

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Appendix 1. The absorption cross section of the NP

The final solution for the scattering, absorption, and extinction cross sections of spherical nanoparticles is as follows [2],

$$ {C_{sca}} = \frac{{2\pi }}{{{k^{2}}}}\sum {(2n + 1)({{\left| {{a_{n}}} \right|}^{2}} + {{\left| {{b_{n}}} \right|}^{2}})}, $$
$$ {C_{ext}} = \frac{{2\pi }}{{{k^{2}}}}\sum {(2n + 1){\text{Re}} ({a_{n}} + {b_{n}})} , $$
$$ {C_{abs}} = {C_{ext}} - {C_{sca}}. $$

Where k is wave vector and an and bn represent the scattering coefficients in terms of the Ricati-Bessel cylindrical functions.

Appendix 2. Expression of the Navier Stokes equations in spherical coordinates

It is assumed that internal energy (eb) changes due to the bubble temperature variations, fluid evaporation, and the motion of the bubble wall, which is described by

$$ \rho_{b} de_{b} = \rho_{b} c_{b}dT_{b} + l(\boldsymbol{\nabla} .\boldsymbol{v})dt - \boldsymbol{\varPi}:(\nabla v)dt, $$
$$ T_{b}{\left( {\frac{{\partial P_{b}}}{{\partial T_{b}}}} \right)_{\rho} }\varDelta V_{b} = L, $$
$$ T_{b}{\left( {\frac{{\partial P_{b}}}{{\partial T_{b}}}} \right)_{\rho} } = l. $$

where L, l, and Vb indicate the evaporation latent heat, clapyron coefficient, and the bubble volume, respectively. ∇v is the “velocity gradient” tensor defined by (∇v)ij = jvi and “:” is the dyadic product as π : (∇v) = πij(∇v)ij. Equation 14 is the Clausius-Clapeyron relationship. Now, if energy conservation equation is written in terms of internal energy and is combined with Eq. 13, the following equation based on the bubble temperature is obtained,

$$ \begin{array}{@{}rcl@{}} \rho_{b} c_{b}\!\left( \frac{{\partial {T_{b}}}}{{\partial t}} + (\boldsymbol{v}.\boldsymbol{\nabla} )T_{b}\!\right) \!&=& - l(\boldsymbol{\nabla} .\boldsymbol{v}) + \boldsymbol{\nabla} .(k_{b}\boldsymbol{\nabla} T_{b})\\ &&+ \boldsymbol{\sigma} \!:\!(\nabla v) + 4\pi {{R_{b}^{2}}}G_{pb}({T_{p}} - T_{b}).\\ \end{array} $$

Regarding Newtonian flow, σ is written as follows,

$$ {\boldsymbol{\sigma}_{ij}} = \eta ({\partial_{i}}{v_{j}} + {\partial_{j}}{v_{i}} + (\boldsymbol{\nabla} .\boldsymbol{v}){\delta_{ij}}). $$

η indicates the layer viscosity, and i and j are indices in Cartesian coordinates. π is used to model a non-homogeneous environment such as the bubble-water system with limited surface tension,

$$ {\boldsymbol{\varPi}_{ij}} = \eta \left( P_{b} - W\rho {\nabla^{2}}\rho - \frac{W}{2}{\left| {\boldsymbol{\nabla} \rho } \right|^{2}}\right){\delta_{ij}} + W{\partial_{i}}\rho {\partial_{j}}\rho. $$

Where W is assumed to be a constant. The first term represents the pressure of Van der Waals and the following terms are related to the surface tension at the bubble-water interface. The momentum and energy conservation equations take a simpler form when the problem is considered in spherical coordinates and in the radial direction.

$$ \frac{{\partial {v}}}{{\partial t}} = - v\frac{{\partial {v}}}{{\partial r}} + W(\boldsymbol{\nabla} ({\nabla^{2}}\rho_{b} ))_{r} + \frac{{3\eta }}{\rho_{b} }\frac{{\partial^{2}v}}{{\partial r}} + \frac{{6\eta }}{{r\rho_{b} }}\frac{{\partial {v}}}{{\partial r}}. $$
$$ \begin{array}{@{}rcl@{}} \rho_{b} {c_{b}}\!\left( \!\frac{{\partial {T_{b}}}}{{\partial t}} + v\frac{{\partial {T_{b}}}}{{\partial r}}\!\right) &=& - l(r)\frac{{\partial {v}}}{{\partial r}} + (\boldsymbol{\nabla} .(k_{b}\boldsymbol{\nabla} T_{b}))_{r}\\ &&+ 2\eta {\left( \frac{{\partial {v}}}{{\partial r}}\right)^{2}} \!+ 4\pi {{R_{b}^{2}}}G_{pb}({T_{p}} - {T_{b}}).\\ \end{array} $$

In the case of nanosecond pulses, since the slope of the density variations at the bubble-water interface is not sharp, the second term on the right-hand side of Eq. 19 can be ignored. Of course, these equations are also applied to the water medium.

Appendix 3. Parameters value

The values of the gold NP parameters used in the calculations are given in Table 2 and values of the water parameters used in the calculations are determined in saturation state, which are dependent on temperature [33] (http://webbook.nist.gov/chemistry/fluid/). To obtain the bubble parameters such as thermal conductivity, specific heat capacity, and viscosity, a linear relationship is formed between the bubble density and these parameters, which is as follows [16],

$$ X = {X_{vap}} + \frac{{\rho_{b} (r) - {\rho_{vap}}}}{{{\rho_{liq}} - {\rho_{vap}}}}({X_{liq}} - {X_{vap}}). $$

The lower indices liq and vap refer to the parameters in the saturation state of liquid and vapor at a given temperature, respectively. The values of the parameters are indicated by the X approach to the saturated parameters when the bubble density approaches to the saturated density. If the temperature of the bubble exceeds the critical temperature, the values of the parameters at critical temperature are used. The vapor latent heat is much less than the water latent heat, and the velocity of the field in the water is considerably lower than in the vapor. Therefore, the first term of Eq. 20 is significant just at the bubble-water interface. The coefficients of the Van der Waals equation of state are obtained by the following relations,

$$ \frac{a}{{27{b^{2}}}} = 22\quad \text{MPa}, $$
$$ \frac{1}{{3b}} = 322\quad \text{kgm}^{-3}, $$
$$ \frac{{8a}}{{27b{K_{B}}}} = 647.3\quad \mathrm{K}. $$
Table 2 The value of gold parameters

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Movahedinejad, H., Nadjari, H. Temperature Variations of Gold Nanoparticle and Dynamics of Plasmonic Bubble in Water Under Nanosecond Pulsed Laser. Plasmonics 15, 631–638 (2020). https://doi.org/10.1007/s11468-019-01055-z

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  • Gold nanoparticle
  • Nano bubble
  • Hydrodynamic equations
  • Nanoparticle heating