Threshold for Vapor Nanobubble Generation Around Plasmonic

Subscriber access provided by UNIV OF NEWCASTLE

Article

Threshold for Vapor Nanobubble Generation Around Plasmonic Nanoparticles Julien Lombard, Thierry Biben, and Samy Merabia J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b01854 • Publication Date (Web): 15 Jun 2017 Downloaded from http://pubs.acs.org on June 15, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Threshold for Vapor Nanobubble Generation Around Plasmonic Nanoparticles Julien Lombard, Thierry Biben, and Samy Merabia∗ Univ Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622, VILLEURBANNE, France E-mail: [email protected]

∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

June 14, 2017 Abstract Vapor nanobubbles are transient bubbles that are generated by plasmonic nanoparticles illuminated by a pulsed laser, and which have been proposed for cancer therapy. Their physical properties are however poorly understood. In this article, we discuss the conditions of appearance of these nanobubbles, on the basis of a hydrodynamics phase field model. In particular, we critically assess the role of the Laplace pressure which was invoked to control the onset of nanobubble production. We clearly demonstrate that capillary effects have only a mild effect on the process of nanoscale vaporization. We also characterize the threshold of nanoscale boiling under different conditions of nanoparticle size and contact angle. We conclude that a very thin shell of liquid water should be brought at the spinodal temperature Tspin ≃ 550 K, which gives upper bounds for the shell assumption of Katayama et al. 2 who consider that a finite volume should be heated at Tspin to create a nanobubble. The existence of a finite thermal resistance at the interface between the particle and water controls the vaporization kinetics, and severly delays vapor nanobubble generation in the vicinity of the corresponding threshold. Finally, we compare the predictions of the hydrodynamic model to available experimental data, corresponding to respectively nanoseconds and femtosecond pulses. The hydrodynamic simulations are in good agreement with the experimental results of Siems et al. 1 and Katayama et al., 2 provided the possibility of gold nanoparticle melting is taken into account. All these considerations help in building a simple thermal diffusive model, that may reproduce both the threshold and the kinetics of nanobubble generation, depending on the nanoparticle size and the laser pulse duration, without any fitting parameter.

Introduction. Nanobubbles have attracted the attention of scientists in different contexts. On the one hand, gas bubbles having nanoscale dimensions have been reported at hydrophobic inter2


Page 2 of 46

Page 3 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


faces, mainly by atomic force microscopy (AFM). 3 On the other hand, nanobubbles have been also produced by photothermal conversion, involving mainly laser heated plasmonic nanoparticles. 1,4,5 The vapor nanobubbles generated by photothermal means have potential applications in cancer therapy, 6–8 photoacoustic imaging 9 or solar energy conversion. 10 To make a distinction between the very different nature of these two types of nanobubbles, we shall call ”gas nanobubbles” the bubbles that contain a non-condensable gas, like the ones generated by hydrophobic surfaces, and ”vapor nanobubbles” the photothermally induced nanobubbles that we shall investigate below. Apart from the difference in the mode of production of these two types of bubbles, their characteristic size and lifetime differ significantly : gas nanobubbles observed at hydrophobic interfaces are almost flat, owing to the small contact angle of the vapor on the surface in this situation. As a result, only one dimension, the bubble thickness is nanometric, the bubble extension reaching microns. As for the lifetime, the gas nanobubbles have been shown to be stable for days, and it is thought that they dissolve in case of gas undersaturation or no pinning of the interface. 11,12 An extended discussion on the stability of these particular nanobubbles may be found in. 3 At the other extreme, vapor nanobubbles generated by hot nanoparticles are spherical, and in contrast to the gas nanobubbles , they are transient and have a lifetime of a few nanoseconds typically, when the duration of the laser pulse used for their generation is lower than 10ns. The dynamics of a vapor bubble in a liquid bath is a notoriously difficult subject (see 13 for a recent review). The vapor nanobubbles that we shall discuss here differ from the usual vapor bubbles nucleated in a heated liquid by several aspects: first the liquid bath is at room temperature, its temperature is consequently below the saturation temperature. This first difference has strong consequences on the bubble growth since the bath cannot sustain vaporization, and tends to condensate the gas on the contrary. The usual diffusion limited growth regime described in 13 does not exist here, and the bubbles are condamned to collapse after a finite time. A mechanism discussed in a previous article 14 may help the vapor nanobubbles to reach a micron-scale size by heating the interface locally: the direct

3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 46

ballistic energy transfer between the surface of the nanoparticle and the bubble interface that promotes vaporization, nevertheless the bubbles always collapse after some time. Of course after the collapse, the nanoparticle may generate other vapor nanobubbles, as long as it is hot enough. Interestingly, the short life time of these bubbles garantees that they are mostly made of vapor, when the nanoparticle survives the experiment. Indeed, the possible presence of a non-condensable gas (like air) in the liquid bath has been shown

15

to dramatically change

the growth dynamics: it stabilizes the bubbles at long times by mass diffusion of the gas into the bubble. The time scale associated to this process (0.1s reported in 15 ) is however much larger than the life time of the bubbles we consider here (nanoseconds), consequently we neglect it, but this is an interesting mechanism that may help to promote the formation of large bubbles. The second important difference that we shall discuss in detail below is the production mechanism that differs from the usual nucleation-growth scenario. For photothermal vapor nanobubbles, the difficulty in building criteria for the nanobubble production is partly explained by the fact that different groups have reported different laser power thresholds. 1,2,5 The experimental results for the threshold are commonly compared with the minimal energy to cross the spinodal line in hot liquid water around 550 K, in the vicinity of the nanoparticle surface. While Siems et al. conclude that vapor nanobubble are produced when the water temperature at the nanoparticle surface surpasses the spinodal temperature, 1 Katayama et al. claim that at least a 10 nm thick water layer should be heated up in order for a bubble to be created. 2 Conversely, Lukianova-Hleb et al. found fluence thresholds one order of magnitude larger than the threshold corresponding to spinodal temperature, this discrepancy being attributed to the large Laplace pressure that needs to be overcome at the nanoscale. 5 Further insight may be obtained by numerical modeling of the process leading to vapor bubble formation. Despite a decade of experimental investigations, it is only recently that the full phase change problem has been modeled, maybe because of the need to run intensive molecular dynamics simulations or to resort to mesoscopic models in order to tackle the almost micronic length scales involved in the problem. Nevertheless,

4


Page 5 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


all the simulations up to date, have concluded that vapor nanobubble appear when the fluid in the vicinity of the nanoparticle is brought to the spinodal temperature. 16,17 These considerations bring us to a couple of fundamental questions : if spinodal crossing describes the vapor nanobubble generation, can we model the phase change process with a simple thermal model, involving only thermal diffusion and Kapitza resistance ? What is the relevance of hydrodynamics in the process ? What is the role of the Laplace pressure in the formation of the bubbles ? Is it minor, as claimed by some authors, 18 or does it influence the vaporization threshold ? Where does vaporization occur in the steep temperature gradient created by the hot nanoparticles ? Is it superficial as claimed by some authors or should a thick shell of water molecules be brought to the spinodal temperature 550 K ? Apart from these fundamental questions, there are important practical consequences in the possibility to model the relevant processes using a set of simple heat transfer equations, that may be solved either with a commercial software, or writing a simple finite difference code. This would help in the use of simple numerical tools having a relatively fast speed of execution as compared with hydrodynamics codes. In this article, we analyze the generation of vapor nanobubbles surrounding hot nanoparticles using simulations of a free energy hydrodynamic model. We concentrate on irradiation conditions close to the nanoparticle surface plasmon resonance (SPR) and neglect any near field radiation effect. 19,20 We conclude the relevance of spinodal crossing to drive vapor nanobubble formation, at a finite distance between 1 and 2 nanometers from the particle surface. The thermal boundary resistance at the nanoparticle interface is shown to control the vaporization kinetics, and may delay the formation of a vapor layer so as to optimize the energy transfer to the fluid. We also compare our numerical findings with available experimental investigations regarding the threshold for vapor nanobubble production. In this respect, we comment the role of the Laplace pressure in the vaporization process.

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 46

This article is structured as follows : First, we remind the equations of the model, and discuss their physical meanings. In a second section, we propose a simple criterion for vapor nanobubble production and discuss the opportunity to build a simplified thermal model that reproduces the hydrodynamic results. We also compute the pressure inside the bubble at the moment of its formation and discuss the role of the Laplace pressure. In a third section, we compare our simulation findings with experimental data reporting the fluence thresholds as a function of the particle size. We conclude by discussing the important ingredients to build a simple thermal model that may predict vapor nanobubble generation.

Motivations: sketch of the system studied We briefly depict in this section the physical situation that we will adress theoretically. A nanoparticle having a radius Rnp > 4 nm, is surrounded by liquid water, and the whole system is initially at thermal equilibrium at room temperature. We break this equilibrium by heating up the particle on a picosecond timescale at a temperature Tnp through the interaction with a laser pulse of short duration, and with a laser wavelength supposed to be close to the nanoparticle SPR so that cavitation is essentially driven by thermal effects. The article aims at describing the state of the fluid, subsequent to the heating of the particle as represented in Fig. 1. A quantity of prime importance at the nanoscale is the thermal boundary resistance G−1 between the particle and the fluid, the latter being defined from the thermal flux flowing across the solid/water interface and the temperature jump:

G−1 =

Tnp − Ts j

(1)

where Tnp is the nanoparticle temperature, Ts is the fluid temperature at the nanoparticle surface, and j is the heat flux density.

6


Page 7 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Laser pulse

T(r) Tnp Gold

Ts

Tnp R np

R np

Ts

r

Water

Gold Rb

T np

Water

Figure 1: (Color online) Sketch of the system considered: a gold nanoparticle in water, initialy heated up by a strong laser pulse of short duration. The GNP/Laser interaction may result in the formation of a vapor nanobubble surrounding the GNP.

Model The reader interested in the results may skip this section and read directly the section Results. Since the length scales involved in the nanobubble dynamics may reach several microns, molecular dynamics simulations, although offering the flexibility to model the relevant situations, become challenging. 17,21 Here, we resort to a hydrodynamic phase field model based on a free energy density, which can be considered in between molecular dynamics and continuum simulations (Navier-Stokes). Note that the length scale where a continuum description is reasonable has been estimated around 1 nm, 22 which corresponds to the liquid-vapor interfacial width. Therefore, for situations involving length scales larger than a few nanometers, as for the problem of interest here, a continuum description is a good starting point. To reach this milestone, we extend a hydrodynamic scheme based on a diffuse-interface model, 23 and which have been successfully applied to address interfacial heat transport and boiling at nanoscale. 24,25 Our implementation includes the existence of a

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 46

finite thermal boundary resistance at the solid-fluid interface, and which is shown to control the nanobubble formation kinetics.

Fluid model We solve the hydrodynamic equations to describe the dynamics of the fluid around the nanoparticle: ∂ρ + ∇ · (ρv) = 0 ∂t ∂v + v · ∇v = −∇ · (P − D) mρ(~r) ∂t ∂T mρ(~r)cv (~r) + v · ∇T = −l(~r)∇ · v + ∇ · (λ(~r)∇T ) ∂t

+ D : ∇v

(2)

(3)

(4)

where ρ, v, T stand, respectively, for the number density, the velocity field and the temperature field in the fluid; m is the mass of a fluid molecule. cv , l, λ, are the fluid specific heat, Clapeyron coefficient l = T ∂P , and thermal conductivity. Because these quantities may ∂T ρ vary spatially depending on the local thermodynamic state of the fluid, we have emphasized

their spatial dependence. D and P stand respectively for the dissipative stress tensor and pressure tensor. The symbol ”:” represents a dyadic product. While the momentum conservation equation eq. (3) encodes information regarding the local fluid thermodynamics through the pressure tensor, the energy conservation equation (4) accounts for the latent heat of vaporization through the Clapeyron coefficient l. Similarly to the dynamic van der Waals model of Onuki et al., latent heat is released or generated at the liquid/gas interface upon evaporation or condensation. 25 The relation between the Clapeyron coefficient l and the liquid/gas latent heat is briefly derived below. Let us consider a volume of liquid V and assume that we are working at constant temperature, and at the corresponding saturation pressure Psat (T ). Under these conditions, the change of internal 8


Page 9 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


energy is dU = Cv dT + (l − P )dV so that the change of enthalpy accompanying fluid dilation is : dH = ldV, and the change of enthalpy corresponding to the liquid/vapor transition is

∆H =

Z

Vvap

ldV = Vliq

Z

Vvap Vliq

~ · ~v dt Vl∇

where Vliq and Vvap are respectively the initial and final fluid volumes

(5)

1

Using the mass

conservation equation eq. (2), the enthalpy change may be rewritten :

∆H =

Z

ρv ρl

−

dPsat l dρ = Na T (vvap − vliq ) = L 2 ρ dT

(6)

which is exactly the Clausius-Clapeyron relation, 26 with vliq and vvap the liquid and vapor specific volumes. Here, we have used the definition of the Clapeyron coefficient l = T ( dPdTsat )ρ . ~ · ~v in eq. (4) is responsible for the heat Therefore, we have demonstrated that the term l∇ of vaporization. A word is of order concerning the value of the Clapeyron coefficient l, for which we have considered a dependence in temperature in addition to the dependence in density:

l(ρ, T ) = lvap (T ) +

ρ(r) − ρvap (lliq (T ) − lvap (T )) ρliq − ρvap

(7)

where lliq (T ) and lvap (T ) are the temperature-dependent values of the Clapeyron coefficient at a given temperature T for the liquid and vapor of density ρliq and ρvap . It is important to remind that equations (2,3,4) are quite general, and may be derived from the conservation of mass, linear momentum and internal energy. The expression of the reversible terms needs to be provided however by a free energy model. Let us start with the expression of the pressure tensor P appearing in eq. (3). To see how it may relate to the local free energy density, we consider first the situation of a bulk phase at a constant value of the temperature and the density. In this case, the pressure P is a scalar quantity and identifies with the 1

Indeed, we have assumed for simplicity that the fluid is homogeneous, i.e. we neglect any temperature or density gradient, but the generalization to heterogeneous situations is straightforward.

9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 46

thermodynamic pressure :

P =−

∂F ∂V

=ρ T

∂fbulk ∂ρ

T

− fbulk

(8)

where fbulk is the bulk free energy density, whose expression should depend on the model retained to describe the intermolecular interactions in the fluid. For instance, in the van der Waals model the free energy writes : ρΛ3 − 1 − aρ2 fVdW (ρ, T ) = ρkB T ln 1 − ρb

(9)

where a and b are the van der Waals parameters and Λ is the De Broglie wavelength. The corresponding pressure is : PVdW =

ρkB T − aρ2 1 − ρb

(10)

Now, in heterogeneous systems as in the vicinity of surfaces and/or interfaces, the pressure tensor is no longer isotropic, and is not described by a single scalar quantity. For instance, in the vicinity of a flat liquid-vapor interface the normal pressure Pzz is larger than the tangential pressure Pxx = Pyy , and the difference is related to the surface tension : γ = R∞ (Pzz − Pxx )dz. 27 Indeed, for heterogeneous systems the pressure tensor contains terms −∞ which are non-local, and which depend on the local density gradients. These latter terms are responsible for the existence of surface tension and a finite interfacial width. If we model the fluid with a square gradient theory, a common basis of diffuse interface models, 23,27 the total free energy takes the form:

F =

Z h

fbulk (~r) +

V

i w |∇ρ|2 d~r 2

(11)

where fbulk (~r) = fbulk (ρ(~r), T (~r)) and w is a coefficient that penalizes the existence of density

10


Page 11 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


gradients, and which is related to the fluid surface tension through: 27

γ=

√

2w

Z

ρliq ρvap

q fbulk (ρ) − µeq (T )ρ + Peq (T ) dρ

(12)

where ρvap and ρliq are the coexistence vapor and liquid densities on each side of a liquidvapor interface at temperature T . µeq (T ) and Peq (T ) are the chemical potential and pressure of both the liquid and vapor bulk phases at equilibrium at temperature T , as given by the bulk equation of state. Finally, in the square-gradient model, the components of the pressure tensor are given by the so-called Korteweg expression :

Pα,β =

h

Pbulk − wρ△ρ +

i w (∇ρ)2 δαβ + w∂α ρ∂β ρ 2

where Pbulk is the bulk thermodynamic pressure, given by eq. (8), and α, β represent the spatiaml directions x, y, z. Hence, the pressure tensor contains all the information regarding the local thermodynamics of the fluid and the capillary terms. In all the following, we will consider a van der Waals free energy density to describe the fluid thermodynamics, see eq. (9). The parameters a, b and the de Broglie wavelength Λ in eq. (9) are set so as to represent the density of liquid water at 297 K and atmospheric pressure and the critical pressure and temperature. The relation between the parameters in Eq. (9) and the critical parameters are given by Eqs. (13).

a = 22 MPa 27b2 1 = = 322 kg/m3 3b 8a = = 647.3 K 27bkB

Pc = ρc Tc

(13)

The parameter w appearing in the square gradient term, was set so as to match the surface tension γ of water at T = 297 K, see eq. (12) and the value provided in Tab 1. 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 46

Table 1: Thermophysical parameters in the liquid (top row) and in the vapor (bottom row) at 297 K in SI units unless specified. Density2 997.10 2.22 10−2

Cv 3 4.13 1.44

λ 0.606 0.019

η 8.98 10−4 9.9 10−6

l 5.4 108 6881

γ 72.0 10−3

The dissipative stress tensor writes:

Dαβ

2 = η ∂α vβ + ∂β vα − ∇.v δαβ + µ ∇.v δαβ 3

(14)

We assume that the shear and bulk viscosities η and µ involved in (14) are related by : µ ≃ 5η/3 28 as it is the case for hard spheres. The thermophysical and transport coefficients of liquid water and vapor at 297 K and atmospheric pressure are summarized in Table 1, together with the water surface tension γ. Since the density ρ in eq. (2) is a field with large spatial variations, we need to account for the variations of the thermophysical and transport coefficients with the local density. For simplicity, we choose a linear relationship between those parameters and the density. As an example, the local shear viscosity is given by:

η(r) = ηvap +

ρ(r) − ρvap (ηliq − ηvap ) ρliq − ρvap

(15)

where the subscripts vap and liq refer to the bulk values, as given in Table 1 at 297 K.

Nanoparticle We discuss now the interaction of the fluid with the nanoparticle. We account for the wettability of the fluid which sets the density at the GNP surface, the interface resistance for thermal conduction, the continuity of pressure and the no-slip condition for the normal velocity. This gives the following boundary conditions at the fluid/GNP interface:

12


Page 13 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Φ w

(16)

v(r = Rnp ) = 0

(17)

(∇.P(r = Rnp )) .n = 0

(18)

(∇ρ)r=Rnp =

G (∇T )r=Rnp = − (Tnp − Ts ) λ

(19)

where Rnp denotes the nanoparticle radius. The potential Φ in (16) quantifies the wetting of the fluid at the first order. 29 It can be directly related to the fluid/solid contact angle, as expressed by eq. 20 below, and illustrated in fig. 3 of ref. 30 so we chose the value for Φ giving the expected contact angle at 297 K. 1 cos θ = 1 − γ

Z

ρsl ρsv

p 2w (fVdW (ρ) − µcoex ρ + Pcoex ) + Φ dρ

(20)

Equation (17) expresses the fact that the fluid does not penetrate in the nanoparticle, while equation (18) stands for the continuity of pressure at the particle surface, n being the unit vector perpendicular to the GNP surface. Finally, eq. (19) is the equation of continuity for the temperature field, where Ts denotes the fluid temperature at the nanoparticle surface, and G is the thermal boundary conductance. For the metal nanoparticle, the temperature Tnp is assumed to be uniform, a reasonable hypothesis owing to the large conductivity of the metal. This assumption was confirmed in a simulation work for quasi-instantaneous heating of the nanoparticle, and for continuous illumination in the limit of small GNP radii, by. 31 The temporal evolution of the nanoparticle temperature is described by

Vnp Cnp

Π(t/tp ) dTnp = F σnp − Snp φ dt tp φ = G(Tnp − Ts )

13


(21) (22)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

where Snp , Vnp are the nanoparticle surface and volume respectively and Cnp =2500 kJ/m3 K is the gold specific heat. The laser interaction is described by the size-dependent GNP absorption cross-section σnp for a laser wavelength λ = 400 nm as given in ref., 1 the fluence of the laser pulse F , and the gate function Π(t/tp ) = 1 if 0 < t < tp ; 0 otherwise, where tp stands for the duration of the GNP heating.

The parameter tp = 7 ps used here is larger than the pulse duration. Since we consider femtosecond pulses the relevant time for our study is the electron-phonon coupling time as described by, 32 which determines the GNP phonon equilibration time: for t ≥ tp , the GNP lattice has received all the energy initially provided to its electron gas by the laser pulse. The electron interaction with the pulse prior to that coupling is not accounted for, as its characteristic time is much smaller than tp . The last term in (21) is due to the surface heat flux φ pointing towards the fluid. Since in this study we are primarily interested in the threshold for nanobubble formation, we have not considered a ballistic heat flux, which controls the bubble dynamics once formed as shown in our previous studies. 14,30 In all the following, we will consider φ = G(Tnp − Ts ) 21,33–36 where the thermal boundary conductance G characterizing the gold/fluid interface writes G = G0 (1 + cos(θ)) as a function of the contact angle. 37–39 Ts = T (Rnp , t) is the temperature of the fluid in contact with the nanoparticle.

We study particles of radii varying from 2 nm to 50 nm. We consider a spherical symmetry that allows us to focus only on the radial component of the fields. The size of the simulation cell, centered at the position of the GNP centre, is N = 260 nm for Rnp < 50 nm and N = 390 nm for Rnp = 50 nm. We solve the set of equations (2,3,4) using a standard finite differences finite time (FDFT) algorithm with a time step of 0.2 fs and a lattice step of 0.07 nm. The velocity field is calculated on a staggered grid, shifted from the main grid by half a lattice step. Perfectly matched layers are used at the simulation box boundaries to avoid

14


Page 14 of 46

Page 15 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


any spurious reflection of the acoustic waves generated by vaporization, as detailed in. 30 For a system size of one micron the temporal evolution of laser-induced nanobubbles over 1ns requires typically 7 CPU hours using 8 threads with a code parallelized using openmp.

Results: Nanobubble formation. The bubbling criterion The theoretical framework presented above will be used to discuss nanobubble formation. However, it is tempting to compare the numerical findings obtained with the full model to a much simpler one, based on temperature diffusion only. Indeed, we shall see that a simple thermal model is able to give quantitative predictions provided the position of the interface is redefined, and a Kapitza resistance is taken into account. At the nanometer scale an interface is not sharp, the molecular organization around a nanoparticle differs from the bulk and a wetting layer is usually observed at the nanoparticle interface. An illustration of such behavior is already visible in Fig. 2 with our simple Van der Waals approach. As a consequence it is necessary to look a bit apart from the nanoparticle surface (1nm or 2nm inside the fluid phase) to recover bulk-like behaviors that may be included in a more macroscopic approach. We consider in this part how this distance (1nm or 2nm) is obtained. Let us discuss here the conditions of formation of the nanobubbles, as observed in the simulations of the full model. In all the following, we consider that a bubble is formed if there is at least one point in the fluid where the density is below the water critical density ρc = 322 kg/m3 . Fig. 2 displays an example of density and temperature profiles, at the moment when a bubble forms. In the close vicinity of the particle surface (denoted by Rnp on the figure), we can observe a small excess of molecules between Rnp and Rnp + xvap that is due to the wettability of the fluid as described by (16). More precisely, this excess of molecules comes from the attractive nature of the interactions between the surface and the molecules of the fluid, in wetting conditions. This wetting layer vanishes when the contact 15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

angle of the fluid on the surface of the nanoparticle reaches 90o and becomes a depletion layer above this value, i.e. for non-wetting surfaces. The typical extension of this wetting layer is given by the range of the fluid/nanoparticle interactions, and is usually around a nanometer in simple liquids. Such a wetting layer is also observed in molecular dynamics simulations, 17 and is visible even far from the thermodynamic equilibrium when the fluence is not too large as we shall discuss below. An immediate consequence of the presence of this layer is that by definition xvap is positive, even at the fluence threshold and vanishes strictly only for a wetting angle of 90o . The red dashed arrow on the left of Fig. 2 emphasizes the temperature drop at the surface of the nanoparticle due to the Kapitza resistance that delays the energy transfer from the GNP to the fluid. From this snapshot of the temperature and density profiles at the very moment when the bubble appears it is possible to identify not only one distance from the nanoparticle where things happen, but at least three. In Fig. 2 we spotted these three characteristic distances xvap , xspin and xinterf when vaporization occurs, at a time that we denote tvap . The three distances are defined as follows: • xvap = r(ρ = ρc ) − Rnp is the position where the vaporization first occurs, where the local density is below the critical density. • xspin = r(T = Tspin ) − Rnp is the position where the temperature of the fluid matches the spinodal temperature. • xinterf = Rb − Rnp is the distance of the liquid/vapor interface from the GNP surface, where the local density matches the mean density. Strictly speaking, these three distances are different. From Fig. 2 we can see that the interface position is quite close to the spinodal crossing position. Since it is a general feature, we shall restrict the discussion to xvap and xspin in what follows. xvap is the distance that comes naturally from our vaporization criterion, but this quantity requires the knowledge of 16


Page 16 of 46

Page 17 of 46

1600 Density Temperature 1400

1500 Temperature drop at the GNP surface

1200 1000

1000

xinterf

Average density

xspin

T (K)

3

ρ (kg/m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


800 600

500

Tspin

ρc

400

xvap

Rnp

16

18

200

radius (nm)

Figure 2: (Color online) Density and temperature profiles in the fluid surrounding a nanoparticle when a bubble forms. Here the particle radius is Rnp = 15 nm and the laser fluence F = 101.25 J/m2 . The dotted lines show, from top to bottom, the position of the spinodal temperature Tspin , the average density and the critical density ρc . The red dashed arrow on the left emphasizes the temperature drop at the surface of the nanoparticle. In this figure we spotted the position of the minimum density, the position of the interface and that of the spinodal crossing, respectively at a distance xvap , xinterf and xspin from the GNP surface. the density profile close to the nanoparticle, a quantity that is difficult to measure experimentally, and that requires the solution of the full hydrodynamic model from the theoretical point of view. In order to use simplified approaches such as the thermal model presented below, it is necessary to transfer the bubbling criterion obtained with the density to the temperature profile. This is where xspin comes into play, this distance is associated with the spinodal temperature crossing and can be used more conveniently. We can see that the onset of bubbling corresponds to the spinodal temperature crossing not at the interface, but on average roughly 2nm away from it. The two distances xvap and xspin depend on the size of the 17



nanoparticle, the fluence of the laser and the wettability of the nanoparticle. When changing the nanoparticle radius, and changing the fluence it is possible to follow the variation of xvap and xspin as plotted in Fig. 3.

2.5

2

x (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 46

1.5

spinodal crossing 1

vaporization 0.5

100

1000

100 nm 50 nm 30 nm 25 nm 20 nm 15 nm 10 nm 5 nm 10000

2

Fluence (J/m ) Figure 3: (Color online) Position of the minimum of density xvap (open symbols) and position where the spinodal is crossed xspin (full symbols), as a function of the laser fluence, and for different GNP sizes. Here, the positions are calculated when a nanobubble is generated. The open symbols (lower curve) corresponds to xvap , and we can see that this quantity follows a master curve, decreasing with the fluence. xvap is only weakly affected by the radius of the particle as proved by the collapse of the curves for different values of the radius on a single master curve. This is an indication that xvap is essentially controled by the nature of the liquid-nanoparticle interface (wetting effect) that is the same for all the particles considered in Fig. 3 (wetting angle θ = 50o ). Of course, when the radius of the nanoparticle is changed, the threshold fluence is modified and a variation of xvap with the radius is still observed as a consequence of its sensitivity to fluence. The reason why xvap decreases with fluence is more complex, it is essentially due to the vaporization time tvap that 18


Page 19 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


decreases with fluence, giving less time to the density profile to reach a local equilibrium. As a consequence, the rapid drop of the density at the surface of the particle prevents a wetting layer to form, and xvap thus tends to zero at large fluences. Let us now discuss the variations of xspin . For each value of Rnp considered xspin corresponds to the full symbols in Fig. 3. For each radius there is a fluence threshold Fthr below which no bubble is observed. As soon as the threshold is overpassed xspin increases rapidly to reach a maximum and decays slowly, as visible for Rnp = 10 nm for example. The threshold fluences correspond to the lowest values of xspin . It is interesting to remark that the threshold values for xspin correspond quite well to the values of xvap + 0.9 nm. This indicates that xvap and xspin behave in the same way at the onset of bubbling. Using a criterion on xspin or xvap to get the fluence threshold is thus equivalent, and we can see in Fig. 3 that the minimum values for xspin are between 1 and 2 nm, at least for nanoparticle radii larger than 5 nm. Above the threshold, the value of xspin can be obtained from Fig. 3, and we can see that they vary in the range 2 ± 1 nm. To emphasize the physical relevance of these length scales, we show in Fig. 4 a plot of the temperature measured in the fluid phase when a bubble is produced, at a typical distance 1nm away from the nanoparticle surface (intermediate between xspin and xvap ). It is interesting to see that these temperature curves collapse on two master curves corresponding respectively to the formation of the first bubbles, and to the formation of the secondary ones. The first bubble is a result of the fast temperature increase of the GNP, and of the energy transfer to the fluid. After the first bubble is produced, the insulating vapor layer that is formed around the GNP prevents quite efficiently its cooling. After the collapse of the first bubble, the temperature of the GNP is usually large enough to generate a second bubble, and possibly a full series of bubbles. The master curves plotted in Fig. 4 show that the formation mechanism is universal in the sense that it does not depend on the GNP size, but there is a strong difference between the first and the secondary bubbles. Since this point was discussed in a previous paper, 14 we shall only mention here that the formation of the first bubble interferes with the formation of a pressure wave, requirering more energy, while the

19



1200

1000

2 nm 5 nm 10 nm 15 nm 20 nm 30 nm 50 nm

first bubbles

T(K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 46

0.9

T/Tc

800

Tspin

0.85

secondary bubbles 1

600

Tspin

2

4

6

F/Fthr

8

1.05

10

F/Fthr Figure 4: (Color online) Water temperature at a distance 1 nm from the GNP, calculated when the nanobubble is generated, as a function of the laser fluence. We have considered different GNP sizes, and separated the first generated bubble and the secondary bubble if any. Tc = 647.3 K. secondary bubbles are easier to form, this explains the lower temperature observed for the secondary bubbles. Finally, the two distances xvap and xspin depend also on the nanoparticle contact angle, as shown in Fig. 5. The two distances are decreasing functions of the wetting angle, that can be understood by the lower attraction exerted by the particle: the wetting layer is less pronounced for large values of the wetting angle. The wetting layer even disappears for the neutral wetting condition θ = 90o , as already discussed, with the consequence that xvap ≡ 0 for that particular value of the wetting angle. Again, there is a weak variation with the radius of the nanoparticle that comes from the variation of the threshold fluence. The insert shows the variation of xspin and xinterf with the liquid-vapor surface tension γ, and we can note a linear increase of the two distances with γ that comes from the thickening of the interface 20


Page 21 of 46

xspin Rnp=10 nm

2.5

xvap xspin xinterf

3

xspin, Rnp=50 nm xvap, Rnp=50 nm xvap, Rnp =10 nm

x (nm)

3

x (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 1 0 0

2

50 100 150 200 250 2

γ (mJ/m )

1.5 1 0.5 0 0

20

40

θ (degree)

60

80

Figure 5: (color online) xvap and xspin at the threshold fluence as a function of the contact angle θ for Rnp = 10nm and Rnp = 50nm. The dashed lines are only here to guide the eye. The inset shows the distances xvap , xspin and xinterf in the case Rnp = 10nm and θ = 90o as a function of the surface tension of the fluid, in this very special case xvap is always 0.

21



when γ is increased (there is in fact a linear relation between γ and the interface thickness within our model, 27 that explains the linear variation of xinterf with γ). As previously said, xspin and xinterf follow the same trend. At this point, we can conclude the relevance of a thermodynamic criterion for explosive boiling through the crossing of the spinodal temperature, at a distance xspin away from the nanoparticle surface. This distance is given by Fig. 3 for various nanoparticles diameters and laser fluences. We shall see bellow that this value can be safely taken between 1 and 2 nm for practical applications using the simplified thermal model.

The simplified thermal model hydrodynamic model thermal model 2

Fluence threshold (J/m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 46

100

2 nm 1 nm surface

0

10

20

30

40

50

60

GNP radius (nm) Figure 6: (Color online) Fluence threshold for vaporisation, as a function of the GNP diameter (symbols) as obtained with the full model. The red dashed show the threshold predicted by the thermal model eqs.(23), corresponding to spinodal crossing at a distance 0, 1 and 2 nm from the GNP surface. The wetting angle is here θ = 50o . The full model used above is interesting because it gives a detailled description of bubbling, from the beginning of the nanoparticle heating to the end of the bubbling sequence. 22


Page 23 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


However, it requires the numerical solution of complex evolution equations that restricts its use to research purposes. On the contrary, a simple thermal model based on temperature diffusion in the liquid phase is easy to solve numerically thanks to present day softwares, semi-analytical expressions are also available in the literature 1 that can be used as well. This model consists in solving the classical heat diffusion equations in a homogeneous fluid, in the vicinity of the GNP: mρcv

∂Tf = λ∆Tf ∂t

(23)

with the kapitza boundary condition at the nanoparticle surface:

φ = G(Tnp − Ts )

(24)

where Tnp is the nanoparticle temperature and Ts the temperature of the fluid in contact with the particle, like in the previous model φ is the thermal flux at the nanoparticlefluid interface. The diffusion equation (23) assumes constant thermodynamic and transport coefficients. As a consequence, the fluid is assumed to remain homogeneous (no bubble) so the thermal model can strictly speaking only mimick the temperature evolution in the fluid before a bubble is produced. Nevertheless, we shall see that this simple approach can give quantitative estimations of the bubbling threshold, and can provide an understanding of the bubbling times if we combine it with the bubbling criterion introduced in the previous part. The thermal model presented here further assumes that the laser pulse instantaneously heats up the nanoparticle at a temperature Tp (F ) that depends on the laser fluence F through the relation: Tp = T0 +

F σnp tp Vnp Cnp

where T0 is the ambiant temperature (297 K) and the other notations correspond to (21). Solution of this model gives the temperature profile in the liquid phase before a bubble is formed, if any. As discussed previously, the threshold fluence for bubble formation corresponds to the 23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

minimal fluence Fmin that leads to the crossing of the spinodal line, at a distance xspin from the GNP surface. Fig. 6 compares this estimation to the results of the thermohydrodynamic simulations for different nanoparticle sizes. It is concluded that the onset of vaporization may be understood in terms of spinodal crossing at a distance varying between 1 nm for the smallest particles to 2 nm for the largest GNPs considered here. This confirms the relevance of a spinodal temperature in the vaporization process, provided that one does not identify the distance xspin where the spinodal temperature is crossed, with xvap < xspin the distance where vaporization first sets in. The spinodal criterion may equally allow to understand the kinetics of vaporization following the laser pulse, as we will see now. Indeed, solving the thermal model eqs. (23) permits to predict the vaporization time, which is defined simply here as the time necessary for the fluid to cross the spinodal at a distance xspin from the GNP surface. The resulting kinetics is reproduced in Fig. 7, together with the simulation vaporization times. Again, the spinodal criterion is found to predict the kinetics of vapor production reasonably well. At this stage it is interesting to discuss the value of xspin used in the simplified model that best fits the full hydrodynamic data. From Fig. 3 we could expect a value between ≃ 1.3 nm close to the threshold up to ≃ 2 nm far from it, and the best fit is obtained for the intermediate value xspin = 1.5 nm. It is important to note that the simplified thermal model forgets many physical effects that may play a role in the kinetics of the bubble formation; one of them is the expansion of the liquid due to heating that affects the local thermal transport, and thus the dynamics . For this reason, the simplified model is not expected to provide very accurate values for the vaporization time. It is interesting to see that, indeed, this simplified approach gives quite good results with a value of xspin that corresponds to the average value expected from Fig. 3. This being said, Fig 7 clearly shows that the vapor production kinetics may be very slow in the vicinity of the fluence threshold. In particular, the vaporization time may become quite longer than the characteristic heat diffusion time τdiff = x2interf /Dth across the interface, and

24


Page 24 of 46

Page 25 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


with Dth = λ/mρcv the heat diffusivity of liquid water. To show it more easily, it is interesting p to introduce the diffusion radius Rdiff = Rnp + Dth tvap that measures the distance from the center of the particle reached by the thermal spot when a bubble is produced. The variations of Rdiff with the laser fluence are plotted in Fig. 8. This figure shows that Rdiff precisely coincide with the position of the liquid-vapor interface at the bubble formation, somehow, it tells us that heat is essentially used to produce the interface. Close to the fluence threshold on the contrary, Fig. 8 shows that the diffusion radius is larger than the interface, so a non negligible part of the energy is transfered to the liquid itself, and is not used to produce the bubble. To control the efficiency of bubble production, it is important to understand the origin of this effect. The existence of these long vaporization times have been already observed in 16 and attributed to the presence of a finite thermal resistance at the interface between the GNP and fluid water. Indeed, running simulations with a vanishing boundary resistance leads to a vaporization kinetics one order of magnitude faster in the vicinity of the threshold as shown in Fig. 7. In the absence of boundary resistance the vaporization times remain comparable with τdiff . For a finite resistance, the local fluid heating subsequent to the initial heating of the GNP is described by an inverse Laplace transform. 40 However the resulting formula is not amenable to simple qualitative interpretation and in particular the effect of the thermal boundary resistance on the kinetics of heating may not be simply discussed. However, it is worth mentionning the solution of the corresponding unidimensional problem. 40 The fluid heating reads: (Tp − T0 )h Tf (x) = T0 + β−α

p x 2 exp αx + Dth tα erf c √ + α Dth t 2 Dth t p x 2 + β Dth t − exp βx + Dth tβ erf c √ 2 Dth t

(25)

where x > 0 is the distance to the interface, Tp is the initial solid temperature, h = G/λ and Dth is the heat diffusivity of liquid water. The two wavevectors α and β are solutions of

25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

q 2 + hq + hh′ = 0 with h′ = ρf cf /cs is the ratio of the specific heats of the fluid and the solid. This formula introduces two time scales 1/(Dth α2 ) and 1/(Dth β 2 ). In fact, these two time scales are typically larger than the diffusion time τdiff which may make the heating process very slow, especially in the vicinity of the fluence threshold. The same argument should hold in the spherical geometry relevant to the nanoparticle heating. This is materialized in Fig. 9 which displays the heating of the fluid at a distance x = 2 nm from the GNP surface, for different fluence levels. For the lowest level shown in this figure, the temperature of liquid water never crosses the spinodal and no vapor bubble is generated. For higher fluences, the local fluid temperature may overpass the spinodal temperature and vaporization proceeds after a finite time. Clearly from Fig. 9, if the fluence is high enough, vaporization is a fast process which occurs on a time scale smaller or comparable with τdiff . On the contrary, at the threshold the appearance time may be several times longer than τdiff as already seen with the full model in Fig. 7. The slow vaporization is attributed here to the very flat shape of the function describing the heating of the fluid close to its maximum. Figure 9 also clearly shows that in the absence of interfacial thermal resistance, the kinetics of vaporization is several times faster, comparable with τdiff . In conclusion, we stressed out the relevance of spinodal crossing in the process leading to nanobubble generation. This allows to understand the variation of the laser fluence threshold with the GNP radius, and contact angle. Capillary effects seem to play a minor role, or at least can not enhance significantly the onset of vaporization. Thus, we conclude that vaporization is a local process driven by the instability of the liquid. We also conclude that strictly speaking the location of the vaporization (xvap ) and that of the spinodal crossing (xspin ) do not match, although these two distances remain confined in the immediate vicinity of the nanoparticle, at a distance smaller than 2 nm.

26


Page 26 of 46

Page 27 of 46

120

Vaporization time (ps)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


100 2 nm

80 60 1 nm 1.5 nm

40 Diffusion time over 2 nm

20 1.5 nm G→∞

Vapor appearance 200 threshold Fluence

2

400

(J/m )

Figure 7: (Color online) Vaporization time tvap as a function of the laser fluence. Symbols: simulation data. Lines: predictions of the thermal model eqs. (23) with a finite thermal interface resistance. The dashed line stands for a simulation with a 10 nm GNP and an infinite interface conductance.

Role of the Laplace pressure A criterion that must be fulfilled for a bubble to grow is that its internal pressure must exceed the Laplace pressure applied by the liquid-vapor interface. The Laplace pressure is directly related to the surface tension γ of the liquid-vapor interface and to the radius Rb of the bubble : ∆PLaplace = 2γ/Rb . It is maximum when the bubble is just produced (its radius is minimal Rb ≃ Rnp ), and a natural question is to know if the Laplace pressure plays a role or not at the bubbling threshold. To investigate this point, we first plot in Fig. 10 the vapor pressure inside the bubble, at the very moment of its formation thanks to the full hydrodynamic model. We can see on this figure that the pressure is minimal at the fluence threshold Fthr , and increases monotonously with the laser fluence F , which makes sense since a larger fluence brings more energy to the bubbles. 27



14

Characteristic diffusion radius Rdiff Appearance radius Rapp Liquid side of the interface Vapor side of the interface

13

R(nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 46

12

Interface thickness linterf 11

10

100

200

300

400

500

2

Fluence (J/m ) Figure 8: (Color online) Position where the nanobubble appears (xinterf , black symbols) compared with the thermal diffusion radius (red symbols, see text for definition) as a function of the laser fluence. The blue symbols indicate the 10-90 interface thickness, i.e. the positions where the local density is respectively 10% of the bulk liquid value (triangles up) and 90% (triangles down). The GNP radius is 10 nm here. At the fluence threshold, we can wonder whether the Laplace pressure controls the nanobubble generation. To answer this question we show in Fig. 11 a plot of the pressure inside the bubble, at the threshold fluence and at the beginning of its growth (the ’Threshold vapor pressure’ in Fig. 11). It is clearly visible that the threshold vapor pressure is always larger than the Laplace pressure, as expected, but it may play a role for the smallest nanoparticles. Indeed, there is a crossover between two regimes: the large particle regime, where the threshold pressure corresponds to the critical pressure (this indicates that the fluid in contact with the particle follows the saturation curve until it reaches the critical point), and the small particle regime where the Laplace pressure is the limiting phenomenon. The crossover between these two regimes correspond to nanoparticles of radii ≃ 7nm which is 28


Page 29 of 46

1400 1200

T (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Diffusion time over 2 nm

2

Tnp at Fthr

1000 800

F = 160 J/m 2 F = 72.5 J/m 2 F = 72.5 J/m (Tnp) 2

F = 72.3 J/m (G→∞) 2 F = 50 J/m Spinodal temperature

Appearance time at threshold

600 400 0 τ diff

50

100 delay(ps)

150

200

Figure 9: (Color online) Temperature of the fluid water at a distance x = 2 nm from the GNP surface, for different laser fluences, below and above the fluence threshold for vaporization. For the threshold fluence, we have also displayed the temperature of the fluid if we neglect the thermal boundary resistance between the GNP and fluid water (dotted black lines). The dashed green line indicates the crossing of the spinodal, here supposed to be given by Tspin = 0.88Tc . Finally, the purple dashed line shows the temperature of the nanoparticle, in the conditions of vaporisation threshold. The GNP radius is here Rnp = 10 nm. quite small. It can be tempting to conclude that the surface tension has a negligible role for particles larger than 7nm, it is indeed true for very large particles, but its effect, although small, remains visible for particles as large as 50nm as shown in Fig. 12. This sensitivity is due to the variation of the Laplace pressure with γ, that goes closer to the critical pressure when γ is increased. The crossover radius Rc is consequently an increasing function of γ:

Rc = 2γ/Pc

where Pc is the critical pressure of water. Multiplying γ by a factor of 4, as done in Fig. 12 will multiply the crossover radius by the same value (Rc ≃ 28nm). This explains why an 29



100

Pvapor at bubble appearance (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 46

90 80 70 60

Rnp=5nm Rnp=10nm Rnp=15nm Rnp=20nm Rnp=30nm Rnp=50nm

50 40 30 20

1

1.2

1.4

1.6

F /Fthr Figure 10: (Color online) Pressure in the vapor calculated at the nanobubble generation, as a function of the laser fluence normalized by the threshold fluence. Here, we have considered different GNP sizes and the first generated bubbles. effect is still visible for 50nm radius particles.

Comparison with experiments and relevance of nanoparticle melting In this section, we compare the predictions of the hydrodynamic model with available experimental data measuring the threshold for bubble generation. We begin by briefly summarizing the relevant experimental literature. Lin and Kelly were the first to report experimentally the formation of transient microbubbles around laser heated melanosomes. 41 They found a fluence threshold for the microcavitation phenomenon around 0.1 J/cm2 , but in absence of the knowledge of the absorption cross section of the heated pigment particles, we can not deduce the corresponding temperature

30


Page 31 of 46

Threshold vapor pressure 100

∆PLaplace (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


critical pressure 10

1 0

20

40

60

80

100

Rb (nm) Figure 11: (Color online) Laplace pressure (black circles), critical pressure (red dashed line) and initial pressure inside the bubble at the fluence threshold as a function of the nanoparticle size. threshold. Similar melanosome absorbers in water have been considered by Neumann and Brinkmann, 42 who observed bubble oscillations around the absorbers, but they did not provide an estimation of the threshold for bubble production. Hu et al. 43 were the first to estimate a temperature threshold for explosive boiling, using a pump-probe laser configuration probing the absorption of 20 nm radius gold nanoparticles in aqueous solution. Their estimate for the threshold temperature was T = 550 ± 50 K. A couple of years later, Kotaidis et al. explored the possibility to generate nanoscale vapor bubbles around gold nanoparticles in water heated by femtosecond laser pulses. 44 Using a simple thermal model including the effect of the Kapitza resistance, they showed that their data may be interpreted as the crossing of the spinodal at the surface of gold particles, which corresponds to a temperature of 85% of the critical temperature. 1 Note that they also mention the possibility of gold melting, depending on the nanoparticle size. We will come back to this issue below.

31



160 2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

140

Page 32 of 46

Rnp = 10 nm Rnp = 50 nm

120 100 80 60

1

2

γ/γ0

3

4

Figure 12: (color online) sensitivity of the fluence threshold with the surface tension at fixed contact angle θ = 90o . γ0 is water surface tension at room temperature.

32


Page 33 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In 2010, an optical method has been employed to form gold nanoparticle aggregates in solution. 45 The formation of micronic bubbles of water in aqueous solution has been evidenced, and bubble formation has been shown to coincide with the boiling point of water under atmospheric pressure (100◦ C). We interpret the non-relevance of the spinodal temperature in this latter study as due to the large nanoparticle aggregate size (∼ 5 µm) and the long laser exposure time (around 1s) for which standard nucleation can take place. In 2012, Carlson et al. determined the temperature of water in the vicinity of a gold nanodot excited by a CW laser. 46 It was shown that water may be heated beyond the normal boiling point, and remains in a metastable state up to the spinodal temperature, estimated at 594 ± 17 K. In 2013, Fang et al. measured with surface enhanced Raman scattering (SERS) the temperature at 100 nm diameter Au nanoparticles illuminated by a CW laser, and reported a transition when the fluid temperature exceeds 465 K. 47 However, the initial temperature when the bubble starts to form was not determined, and it has been shown that just after vaporization, the temperature in the vapor shell decreases sharply as a result of the huge increase of the thermal resistance following phase change. 30 More recently, Katayama et al. 2 probed the dynamics of nanobubbles around 20−150 nm colloidal gold nanoparticles irradiated by a ps laser pulse. In contrast with previous studies, the authors assume that the threshold temperature to initiate boiling should be reached for a 25 nm diameter thick water shell to interpret their results. In parallel, the Lapotko group measured the threshold of transient nanobubbles around 60 nm gold nanospheres and 250 nm nanoshells illuminated by nanosecond optical excitation. 5 In this latter study, they conclude that the formation of vapor nanoshell requires an energy threshold that exceeds by one order of magnitude the explosive boiling threshold. This huge difference is explained by the effect of surface tension originating from the very large Laplace pressure owing to the nanometric curvature of the nanobubble. 5 In a recent study, the Lapotko group examined the effect of the pulse duration, and reported a 100 fold increase of the nanobubble threshold when the pulse duration is increased from 20 ps to 14 ns. 48

33



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

From this brief review, we conclude that while the initial studies conclude a threshold close to 550 K, recent experimental studies have challenged this original belief. Before turning to the simulation results, we should mention that most of the studies cited above concern pulsed lasers, and the interest for continuous wave heating (CW) came out only recently. 49–51 In particular, Baffou et al. reported the formation of microbubbles around plasmonic nanoparticles, for a water temperature beyond 220◦ C. In this latter study, the bubbles are however made of air, and not water steam, which explains their outstanding stability. 51 In the following, we will compare our simulation results to experimental studies considering single gold nanoparticles illuminated by pulsed lasers. Hence, we will consider respectively the data obtained by the Plech group, 1 the Lapotko group 5 and Hashimotos’s group. 2 Figure 13 compares the fluence threshold as determined in the hydrodynamic model to the experimental data respectively of the Plech group and the Lapotko group. These experimental data correspond to a 0.5 nanosecond laser pulse. We note the large discrepancy between the two sets of data, independently of the nanoparticle size. Strictly speaking, the two experimental studies do not correspond to the same irradiation wavelength, but this can not explain the large discrepancy reported in Fig. ??. In particular, the data from Lukianova et al. 5 can not be interpreted by a simple criterion related to the crossing of the spinodal. These authors interpret the large energy difference with respect to the spinodal crossing as a result of the very large Laplace pressure that has to be overcome to form a nanobubble. We analyzed in the previous section the role of the Laplace pressure, and we saw that it may play a role in the fluence threshold (see fig. 12), but the effect is relatively small for big nanoparticle radius, and can not explain the large discrepancy evidenced in fig. 13. A possible explanation may come from the smallness of the nanobubbles at the threshold, that would not be visible with the optical detection used in the experiment. More probably, another mechanism not included in the present model is relevant in the formation

34


Page 34 of 46

Page 35 of 46

1e+06

2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1e+05

hydro model exp. Siems et al. exp. Lukianova et al.

10000

1000

100

10

1

10

100

GNP radius (nm) Figure 13: (Color online) Comparison between the fluence threshold calculated in the simulations for nanosecond pulses and the experimental values of Siems et al. 1 (blue triangles) and Lukianova et al. 5 (purple symbols). The laser pulse duration here is tp = 0.5 ns. The laser wavelength is λ = 355 nm in Siems et al. and λ = 532 nm in, 5 but the corresponding difference in the nanoparticle absorption cross-section can not explain the discrepancy between the two sets of experimental data. of nanobubbles in their system. Now, let us consider the experimental data by Siems et al. and compare our numerical predictions in the case of femtosecond pulse. The absorption cross-section is taken from, 1 and corresponds to a femtosecond pulse slightly off the SPR (λ = 400 nm). Figure 14 reports the evolution of the threshold fluence as a function of the nanoparticle size for femtosecond pulses. We have also compared the predictions of our model to the experimental data of, 2 which considers a 10 ps laser pulse. Although there is a difference in the pulse duration as compared to the experiments of, 1 the difference in the nanoparticle heating times should be small, and in both cases the heating time is comparable with 10 ps. We clearly see that the hydrodynamic model underestimates the threshold observed ex35



400

exp. Siems et al. exp. Katayama et al. hydro. model hydro. model + melting

2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 46

300

200

bubbles

100

no bubbles 0

10

20

30

40

50

GNP radius (nm) Figure 14: (Color online) Comparison between the fluence threshold calculated in the simulations (black empty symbols) and the experimental values of Siems et al. 1 (full violet squares) and Katayama et al. 2 (green full symbols). We have also indicated the threshold that we estimate if we take into account GNP melting as explained in the text(red symbols). In the experiments, the laser duration is smaller than 10 ps in both cases, λ = 355 nm in 1 and λ = 400 nm in, 2 and the difference in the absorption cross-sections is small. perimentally, and the difference may reach 40 percents. We remark a good agreement between the hydrodynamical model and the experimental result for a 30 nm radius nanoparticle. We will comment this good agreement in the following. Since it was mentionned in 1 the possibility of gold melting, we have quantified the effect of nanoparticle melting in the determination of the threshold fluence. We follow here a simple route to predict the effect of melting, and account for the change in the threshold fluence through the increase Fthr → Fthr + hm Vnp /σnp where hm = 1.24 109 J/m3 is the bulk melting enthalpy of gold, and Vnp the nanoparticle volume. Clearly, the addition of the melting enthalpy improves the agreement with both experimental data, except once again for the 30 nm nanoparticle radius, for which a good agreement was observed without melting. 36


Page 37 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


It is important to stress at this point, that the agreement is remarkable given that there is no free parameter. To further understand the possibility of nanoparticle melting, we have accounted for melting in the thermal model. More precisely, we have numerically solved the temporal evolution :

Vnp Cnp

Π(t/tp ) H(Tnp − Tf ) dTnp = F σnp − Snp Φc − ∆Hf exp(−(t − tf )/τf ) dt tp τf

(26)

where we take Tf = 1200 K for the gold melting temperature, ∆Hf = hm Vnp and τf = 30 ps the characteristic gold melting time as given in. 52 Note that recrystallisation is a much longer process occuring on nanosecond time scales, so we ignore the possibility for the particles to recrystallize after their melting. Figure 15 shows the two fluence thresholds relevant to melting. The lowest one, called the onset of melting, indicates the minimal threshold for the gold nanoparticles to initiate melting. The second threshold corresponds to the threshold for total melting, i.e. the gold nanoparticle is completely molten. From the inspection of fig. 15, we clearly conclude that all the experimental thresholds are beyond the total melting line, except the data of Katayama, for the particular value of a 30 nm radius nanoparticle. This justifies our previous simplified treatment of melting, where we simply add the enthalpy of nanoparticle melting to the fluence threshold found in the hydrodynamic model. The latter figure confirms also the good agreement that we mentioned between the hydrodynamic model and experimental data, for the Rnp = 30 nm nanoparticle. In this particular case, nanoparticle melting may not occur, as a result of the low fluence threshold necessary to generate nanobubbles. Finally, we can come back to the interpretation provided by Katayama et al who relate the deviation from the spinodal criterion to the fact that a 10 nm thick water shell should be heated up to the spinodal temperature 550 K. Our simulations show that to generate nanobubbles, it is sufficient to heat up a very thin shell, with a thickness between 1 and 2 nm depending on the nanoparticle size. Comparing experimental data to

37



our hydrodynamic model points at a different interpretation : the deviations between the data reported experimentally stem from the possibility of the gold nanoparticles to melt. Therefore, an additional energy is not necessary to heat up a finite volume of water, but rather to melt the nanoparticles.

exp. Siems et al. exp. Katayama et al. hydro model Onset of melting Total melting

200

2

Threshold fluence (J/m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 46

150

100

Total melting Partial melting 50

No melting 10

20

30

40

50

60

70

GNP radius (nm) Figure 15: (Color online) Comparison between the fluence threshold calculated in the simulations (black empty symbols) and the experimental values of Siems et al. 1 (full violet squares) and Hashimoto et al. 2 (green full symbols). We have also indicated the threshold that we estimate if we take into account GNP melting (red symbols).

38


Page 39 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Conclusion. In conclusion, we presented a theoretical analysis of nanobubble generation around laser heated nanoparticle, with the aim to rationalize the experimental data regarding the threshold for nanobubble production. Some controversial results have been published recently in the literature, with some studies reporting deviations from the spinodal criterion, i.e. nanobubble appears when water at the nanoparticle surface is brought at the spinodal temperature around 550 K. These deviations have been interpreted, either by the fact that a 10 nm thickness water shell should be brought to 550 K, or by the very large Laplace pressure that has to be overcome in order to form a nanobubble. Our hydrodynamic simulations conclude that nanobubbles appear once water at a distance xspin is heated up at the spinodal temperature. This latter distance differs from the distance where vaporization first occurs, but is always between 1 and 2 nm depending on the nanoparticle size. We also saw that Laplace pressure plays a role for nanoparticle radii less than 10 nm, but for bigger nanoparticles the liquid surface tension has only a small effect. We compared our theoretical results with available experimental data in the literature, regarding nanobubbles generated by both femtosecond and nanosecond pulses. Our theoretical results are below the threshold reported by Lukianova et al., and we clearly rule out the possible role of the Laplace pressure, which is present in our hydrodynamic model. For the case of femtosecond pulses, we showed good agreement between the theoretical model and available data by Katayama et al. and Siems et al., provided we take into account gold nanoparticle melting. The agreement is remarkable given that there is no free parameter. The relevance of melting has been confirmed by a simple thermal model, which shows that the threshold to produce nanobubbles are high enough, so that it is not possible to avoid melting, at least for small nanoparticles. All these considerations may help in building in the future simple thermal models to predict the onset of nanobubble production, taking into account the equation of state of hot water. Another direction of investigation concerns the generation of nanobubbles around large–micronic objects. At large scale, vapor production should occur through the formation of vapor nuclei.

39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

It would be interesting to describe the transition between the regime of spinodal vaporisation studied here and nucleation, using e.g. dissipative particle dynamics simulations. 53

References (1) Siems, A.; Weber, S. A. L.; Boneberg, J.; Plech, A. Thermodynamics of Nanosecond Nanobubble Formation at Laser-Excited Metal Nanoparticles. New Jour. Phys. 2011, 13, 043018. (2) Katayama, T.; Setoura, K.; Werner, D.; Miyasaka, H.; Hashimoto, S. Picosecond-tonanosecond Dynamics of Plasmonic Nanobubbles from Pump-Probe Spectral Mea- surements of Aqueous Colloidal Gold Nanoparticles. Langmuir 2014, 30, 95049513. (3) Lohse, D.; Zhang, X. Surface Nanobubbles and Nanodroplets. Rev. Mod. Phys. 2015, 87, 981. (4) Neumann, O.; Feronti, C.; Neumann, A. D.; Dong, A.; Schell, K.; Lu, B.; Kim, E.; Quinn, M.; Thompson, S.; Grady, N. et al. Compact Solar Autoclave Based on Steam Generation Using Broadband Light-Harvesting Nanoparticles. Proc. Nat. Acad. Sci. 2013, 110, 11677. (5) Lukianova-Hleb, E.; Hu, L. Y.; Latterini, L.; Tarpani, L.; Lee, S.; Drezek, R. A.; Hafner, J.; Lapotko, D. O. Plasmonic Nanobubbles as Transient Vapor Nanobubbles Generated around Plasmonic Nanoparticles. ACS Nano 2010, 4, 2109. (6) Wen, D. Intracellular Hyperthermia: Nanobubbles and their Biomedical Applications. Int. J. Hyperth. 2009, 25, 533. (7) Lukianova-Hleb, E.; Hanna, E.; Hafner, J.; Lapotko, D. O. Tunable Plasmonic Nanobubbles for Cell Theranostics. Nanotechnology 2010, 21, 085102. (8) Lapotko, D. Plasmonic Nanobubbles as Tunable Cellular Probes for Cancer Theranostics. Cancers 2011, 3, 802. 40


Page 40 of 46

Page 41 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(9) Yin, T.; Wang, P.; Zheng, R.; Zheng, B.; Cheng, D.; Zhang, X.; Shuai, X. Nanobubbles for Enhanced Ultrasound Imaging of Tumors. Int. Jour. Nanomedecine 2012, 7, 895– 904. (10) Ni, G.; Miljkovic, N.; Ghasemi, H.; Huang, X.; Boriskina, S. V.; Lin, C. T.; Wang, J. J.; Xu, Y.; Mahfuzur Rahman, M.; Zhang, T. et al. Volumetric Solar Heating of Nanofluids for Direct Vapor Generation. Nano Energy 2015, 17, 290–301. (11) Zhang, X. H.; Quinn, A.; Ducker, W. A. Nanobubbles at the Interface between Water and a Hydrophobic Solid. Langmuir 2008, 24, 4756. (12) Liu, Y.; Zhang, X. Nanobubble Stability Induced by Contact Line Pinning. J. Chem. Phys. 2013, 138, 014706. (13) Prosperetti, A. Vapor Bubbles. Annu. Rev. Fluid Mech. 2017, 49, 221. (14) Lombard, J.; Biben, T.; Merabia, S. Ballistic Heat Transport in Laser Generated NanoBubbles. Nanoscale 2016, 8, 14870–14876. (15) Wang, Y.; Zaytsev, M. E.; The, H. L.; Eijkel, J. C. T.; Zandvliet, H. J. W.; Zhang, X.; Lohse, D. Vapor and Gas-Bubble Growth Dynamics around Laser-Irradiated, WaterImmersed Plasmonic Nanoparticles. ACS Nano 2017, 11, 2045–2051, PMID: 28088847. (16) Lombard, J.; Biben, T.; Merabia, S. Kinetics of Nanobubble Generation Around Overheated Nanoparticles. Phys. Rev. Lett. 2014, 112, 105701. (17) Sasikumar, K.; Keblinski, P. Molecular dynamics investigation of nanoscale cavitation dynamics. J. Chem. Phys. 2014, 141, 234508. (18) Metwally, K.; Mensah, S.; Baffou, G. Fluence Threshold for Photothermal Bubble Generation Using Plasmonic Nanoparticles. J. Phys. Chem. C 2015, 119, 28586–28596. (19) Boulais, E.; Lachaine, R.; Meunier, M. Plasma Mediated Off-Resonance Plasmonic Enhanced Ultrafast Laser-Induced Nanocavitatio. Nanoletters 2012, 12, 4763. 41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(20) Dagallier, A.; Boulais, E.; Boutopoulos, C.; Lachaine, R.; Meunier, M. Multiscale Modeling of Plasmonic Enhanced Energy Transfer and Cavitation around Laser-Excited Nanoparticles. Nanoscale 2017, 9, 3023. (21) Merabia, S.; Keblinski, P.; Joly, L.; Lewis, L.; Barrat, J.-L. Critical Heat Flux around Strongly Heated Nanoparticles. Phys. Rev. E 2009, 79, 021404. (22) Bocquet, L.; Charlaix, E. Nanofluidics, from Bulk to Interfaces. Chem. Soc. Rev. 2010, 39, 1073. (23) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Diffuse-Interface Methods in Fluid Mechanics. Annu. Rev. Fluid Mech. 1998, 30, 139. (24) Onuki, A. Dynamic van der Waals Theory. Phys. Rev. E 2007, 75, 036304. (25) Teshigawara, R.; Onuki, A. Spreading with Evaporation and Condensation in OneComponent Fluids. Phys. Rev. E 2010, 82, 021603. (26) Balian, R. From Microphysics to Macrophysics; Springer, Berlin, 2007. (27) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford University Press, Oxford, 2002. (28) Schweizer, K.; Saltzman, E. J. Activated Hopping, Barrier Fluctuations, and Heterogeneity in Glassy Suspensions and Liquids. J. Phys. Chem. B 2004, 108, 19729. (29) de Gennes, P. G. Wetting: Statics and Dynamics. Rev. Mod. Phys. 1985, 57, 825. (30) Lombard, J.; Biben, T.; Merabia, S. Nanobubbles around Plasmonic Nanoparticles: Thermodynamic Analysis. Phys. Rev. E 2015, 91, 043007. (31) Baffou, G.; Rigneault, H. Femtosecond-Pulsed Optical Heating of Gold Nanoparticles. Phys. Rev. B 2011, 84, 035415.

42


Page 42 of 46

Page 43 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(32) Juve, V.; Scardamaglia, M.; Maioli, P.; Crut, A.; Merabia, S.; Joly, L.; Del Fatti, N.; Vallée, F. Cooling Dynamics and Thermal Interface Resistance of Glass-Embedded Metal Nanoparticles. Phys. Rev. B 2009, 80, 195406. (33) Swartz, E. T.; Pohl, R. O. Thermal Boundary Resistance. Rev. Mod. Phys. 1989, 61, 605–668. (34) Barrat, J.-L.; Chiaruttini, F. Kapitza Resistance at the Liquid—Solid Interface. Mol. Phys. 2003, 101, 1605–1610. (35) Merabia, S.; Keblinski, P.; Joly, L.; Lewis, L. J.; Barrat, J.-L. Critical Heat Flux around Strongly Heated Nanoparticles. Phys. Rev. E 2009, 79, 021404. (36) Cahill, D. G.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R. Nanoscale Thermal Transport. J. Appl. Phys. 2003, 93, 793. (37) Ge, Z. B.; Cahill, D. G.; Braun, P. V. Thermal Conductance of Hydrophilic and Hydrophobic Interfaces. Phys. Rev. Lett. 2006, 96, 186101. (38) Shenoguina, N.; Godawat, R.; Keblinski, P.; Garde, S. How Wetting and Adhesion Affect Thermal Conductance of a Range of Hydrophobic to Hydrophilic Aqueous Interfaces. Phys. Rev. Lett. 2009, 102, 156101. (39) Merabia, S.; Lombard, J.; Alkurdi, A. Importance of Viscoelastic and Interface Bonding Effects in the Thermal Boundary Conductance of SolidWater Interfaces. Int. Jour. Heat Mass Transf. 2016, 100, 287–294. (40) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Oxford Clarendon Press, Oxford, 1959. (41) Lin, C. P.; Kelly, M. W. Cavitation and Acoustic Emission around Laser-Heated Microparticles. App. Phys. Lett. 1998, 72, 2800.

43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(42) Neumann, J.; Brinkmann, R. Self-limited Growth of Laser-Induced Vapor Bubbles around Single Absorbers. Appl. Phys. Lett. 2008, 93, 033901. (43) Hu, M.; Petrova, H.; Hartland, G. V. Investigation of the Properties of Gold Nanoparticles in Aqueous Solution at Extremely High Lattice Temperatures. Chem. Phys. Lett. 2004, 391, 220–225. (44) Kotaidis, V.; Dahmen, C.; von Plessen, G.; Springer, F.; Plech, A. Excitation of Nanoscale Vapor Bubbles at the Surface of Gold Nanoparticles in Water. J. Chem. Phys. 2006, 124, 184702. (45) Liu, Z.; Hung, W. H.; Aykol, M.; Valley, D.; Cronin, S. B. Optical Manipulation of Plasmonic Nanoparticles, Bubble Formation and Patterning of SERS Aggregates. Nanotechnology 2010, 21, 105304. (46) Carlson, M. T.; Green, A. J.; Richardson, H. H. Superheating Water by CW Excitation of Gold Nanodots. Nanoletters 2012, 12, 1534–1537. (47) Fang, Z.; Zhen, Y.-R.; Neumann, O.; Polman, A.; Javier Garcia de Abajoa, F.; Nordlander, P.; Halas, N. J. Evolution of Light-Induced Vapor Generation at a Liquid-Immersed Metallic Nanoparticle. Nanoletters 2013, 13, 1736–1742. (48) Lukianova-Hleb, E.; Volkov, A.; Lapotko, D. O. Laser Pulse Duration is Critical for the Generation of Plasmonic Nanobubbles. Langmuir 2014, 30, 7425–7434. (49) Huehn, D.; Govorov, A.; Gil, P. R.; Parak, W. J. Photostimulated Au Nanoheaters in Polymer and Biological Media: Characterization of Mechanical Destruction and Boiling. Adv. Func. Mat. 2012, 22, 294–303. (50) Hou, L.; Yorulmaz, M.; Verhart, N. R.; Orrit, M. Explosive Formation and Dynamics of Vapor Nanobubbles around a Continuously Heated Gold Nanosphere. New Jour. Phys. 2015, 17, 013050. 44


Page 44 of 46

Page 45 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(51) Baffou, G.; Polleux, J.; Rigneault, H.; Monneret, S. Super-Heating Micro-Bubbble Generation around Plasmonic Nanoparticles. J. Phys. Chem. C 2014, 118, 4890. (52) Inasawa, S.; Sugiyama, M.; Noda, S.; Yamaguchi, Y. Spectroscopic Study of Laser Induced Phase Transition of Gold Nanoparticles on Nanosecond Time Scales and Longer. J. Phys. Chem. B 2006, 110, 3114. (53) Merabia, S.; Pagonabarraga, I. A Mesoscopic Model For (De)Wetting. Eur. Phys. Jour. E 2006, 20, 209–214.

45



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Graphical TOC Entry

46


Page 46 of 46

Threshold for Vapor Nanobubble Generation Around Plasmonic

Recommend Documents