Auxiliary material: Nanobubbles around overheated ...

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Auxiliary material: Nanobubbles around overheated nanoparticles. Julien Lombard, Thierry Biben, and Samy Merabia. Institut Lumi`ere Mati`ere, UMR5306 ...
Auxiliary material: Nanobubbles around overheated nanoparticles Julien Lombard, Thierry Biben, and Samy Merabia Institut Lumi`ere Mati`ere, UMR5306 Universit´e Lyon 1-CNRS, Universit´e de Lyon 69622 Villeurbanne c´edex, France (Dated: November 27, 2013)

1-Fluid model We use a van der Waals free energy density: f (ρ, T ) = kB T log(ρΛ3 /(1 − ρb)) − aρ2 where kB and Λ are the Boltzmann constant and the DeBroglie wavelength, and the parameters a and b define the van der Waals equation of state. The tensor P in eq. 1 of the manuscript accounts for both the thermodynamics and the surface tension of the fluid as described in Ref.[1]: 1 2 Pα,β = (ρ ∂f ∂ρ − f − w(ρ△ρ − 2 (∇ρ) ))δαβ + w∂α ρ∂β ρ is the pressure tensor here expressed as a function of the free energy density f (ρ, T ). The value of the parameter w in P is set so as to describe the liquid-vapor surface tension of water. The dissipative stress tensor in eq. 1 is of the form Dαβ = η (∂α vβ + ∂β vα − 2/3 ∇.v δαβ ) + µ ∇.v δαβ where µ is the bulk viscosity and η is the shear viscosity. Finally, in the ¢energy equation, cv is the specific ¡ last bulk is the thermoelastic Clapeyron heat, l = T ∂P∂T ρ coefficient and the symbol : denotes a dyadic product. 2-Equation for the temperature of the particle The last term in eq. 2 of the manuscript is due to the surface flux flowing towards the fluid. Two situations should be distinguished depending on the phase of the fluid in contact with the nanoparticle: if the fluid is in the liquid state, the outcoming flux is taken to be G(Tnp − Tsurface ) where G is the thermal boundary conductance [2–6] characterizing the gold/fluid interface, and Tsurface = T (R, t) is the fluid temperature in contact with the nanoparticle. When a bubble appears, the nanoparticle is surrounded by a vapor layer and its interface thermal conductance is expected to decrease drastically [7, 8]. As a consequence once the vapor is formed, the conductive heat flux from the nanoparticle to the fluid is small, and the GNP energy is transferred to the fluid through ballistic transport in the vapor nanobubble. This flux depends on the temperature of the particle and that of the fluid at p the liquid-vapor interface. It is 3 /m(T 3/2 − T 3/2 ) where α is modeled as φb = αρs 2kB np G a dimensionless accommodation coefficient set to 0.1, m is the mass of a fluid molecule, ρs is the number density at the particle surface and TG is the temperature of the fluid at the Gibbs position of the liquid-vapor interface. The latter form of the ballistic flux is inspired by the theory of energy transport in a Knudsen layer [9]. 3-Rayleigh-Plesset fitting procedure: Stricly speaking, the adiabatic exponent for water should be between 7/5

and 9/7 as water is a triatomic molecule. However, the dynamic free energy model considers only translational degrees of freedom of water molecules, and discards any rotation of the molecules. As a result, water behaves as a monoatomic vapor and the adiabatic exponent expected is 5/3. Of course, experimental results should be consistent with a different exponent. As for the external pressure in eq. 3, we used (1) as a simple way to account for the pressure evolution outside the bubble as a function of the pressure far from the particle P∞ . ¸ · xcollapse − xmax (t − tmax ) Pe (t) = P∞ xmax + tcollapse − tmax

(1)

This linear expression is consistent with the pressure variation in the fluid close to the liquid-vapor interface as given by the simulations. The difficulty is to know exactly where the pressure should be measured as the interface is not a sharp separation between two phases. Therefore, we chose a global expression and found which set of parameters gives the best results in fitting equation 3. Results are given in table I. It is important to note that whatever coefficients we choose for xcollapse and xmax , provided that those coefficients were relevant, the adiabatic-isotherm combination always gave the best results for the fit. The fitting parameters in table I show that the evolution of the pressure at the bubble’s interface is an important parameter. TABLE I. Parameters for the fit of equation 3. Fluence (J.m−2 ) 162 271

Pimax /P∞

xmax

xcollapse

9.27 6.92

−4.59 −4.56

8.51 4.9

∆Pmax (Pa) 7.06 106 5.86 106

2γ/Rmax (Pa) 1.07 107 9.59 106

4-Diffusive model accounting for GNPs melting: To understand the influence of the GNPs melting, we solve a purely diffusive model that writes: ρcv Vnp Cnp

∂T = ∇ · (λ∇T ) ∂t

(2)

t−τm e(− τm ) ∆Hmelt Tm dTnp = P (t) − Snp φ − Π( ) dt τm Tnp Π(t/tp ) P (t) = F σnp tp

where we used the finite melting time τm =30 ps [10] and the enthalpy of melting ∆Hmelt = Vnp hmelt with hmelt = 1.24 109 J/m3 for gold. Π(Tm /Tnp )=1 only if the temperature of the particle exceeds the melting temperature Tm and 0 otherwise. Solving Eqs. 2 for different radii leads to a melting threshold that accounts for heat diffusion in water, the gold/water thermal boundary resistance and the kinetics of melting.

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