Modelling heat dissipation from plasmonic gold nanoparticles

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Modelling heat dissipation from

plasmonic gold nanoparticles

by

Aniek van Onzenoort

10428585

Supervisor: dr. Elizabeth von Hauff

Second examiner: dr. Greg Stephens

Content: 15 EC

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Abstract

In this project the process of heating golden nanoparticles (NPs) is investigated. The goal: a better understanding of this process with as result more possibilities within nanotechnology. In a MATLAB finite differences model the process of shining a laser beam on a gold NP was visualized. The data from the model is plotted in a graph and compared with experimental data. In order to obtain comparable results, in the model a pulsating laser was used. Then the equilibrium state was modelled and compared to a mathematical model. The mathematical model showed a faster temperature decrease; in the MATLAB model, heat is dissipated at a slower pace. Then one pulse was modelled so both the heating and cooling of the particle could be visualized. The same shape as in the experiment was achieved. It would be interesting to adapt the model to different sizes in order to compare with outcomes of SERS experiments.

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1 Samenvatting voor 6 VWO

In dit onderzoek is de temperatuurstijging en - daling van gouden nanodeeltjes die worden beschenen met een laser gemodelleerd. Nanodeeltjes zijn deeltjes met een grootte tussen de 1 en 100 nanometer. De speciale eigenschappen van deze deeltjes zijn sinds de negende eeuw voor christus bekend. Ze werden toen gebruikt om voorwerpen te laten glimmen of zo te maken dat ze van kleur kunnen veranderen. In 1847 deed Michael Faraday voor het eerst onderzoek naar deze deeltjes. Hij ontdekte dat de kleine gouddeeltjes licht op een bijzondere manier weerkaatsen, anders dan een grote hoeveelheid goud zou doen.

Het verschil tussen een grote hoeveelheid goud en nanodeeltjes, is dat dat de verhouding tussen volume en oppervlak anders is. Dit veroorzaakt naast het weerkaatsen van licht ook andere effecten. Deze nanodeeltjes gedragen zich, als ze met een laser worden beschenen als kleine antennes. Ze versterken namelijk het elektromagnetische veld rond de deeltjes en er komt warmte vrij. Door nanodeeltjes op een bepaalde plek aan te brengen en met een laser te beschijnen, is het hierdoor mogelijk om een specifieke plaats te verwarmen. Bijvoorbeeld om kankercellen die zich te midden van gezonde cellen bevinden te vernietigen.

Gouden nanodeeltjes worden voor verschillende toepassingen gebruikt, het proces van de verwarming en het uitzenden van de warmte is alleen nog niet volledig duidelijk. Er bestaat geen apparatuur waarmee je op deze schaal temperatuur kan meten. De onderzoeksgroep van Elizabeth von Hauff probeert een beter begrip te krijgen door een techniek genaamd SERS te gebruiken om de temperatuur te meten.

In dit onderzoek is geprobeerd om het proces van verwarmen en de warmte uitzenden in een MATLAB te modelleren. Dit door een bestaand model aan te passen en dezelfde fysieke waarden als een uitgevoerd experiment te geven. Als de uitkomsten van het model te vergelijken zijn met de uitkomsten van de metingen biedt dat mogelijkheden voor een beter begrip van het proces. In dit onderzoek is eerst de theorie van de verwarming van de deeltjes onderzocht. Vervolgens waren er experimentele resultaten nodig om het model mee te kunnen vergelijken. Omdat de onderzoeksgroep deze nog niet had, is in de literatuur gezocht naar geslaagd SERS-onderzoek. Dit bleek niet te vinden. Daarom is een ander onderzoek gebruikt om het model mee te vergelijken. Het model in MATLAB produceert een matrix van 100 bij 100 pixels. In het midden bevindt zich een deeltje dat door een laser beschenen wordt. Het model berekent per tijdstap hoeveel warmte het deeltje doorgeeft aan de aangrensende deeltjes en laat zo de verspreiding van de temperatuur zien. Door een grafiek te maken van deze verspreiding, is het model te vergelijken met de warmteverspreiding uit een eerder uitgevoerd experiment, is gecontroleerd of het model goed werkt. Dit bleek het geval te zijn, een volgende stap is nu om het model ter vergelijken met de uitkomsten van SERS-metingen. Hierin zijn onder andere de volgende variaties in het model mogelijk: verschillende vormen, maten en aantal deeltjes.

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2 Introduction

The special properties of plasmonic particles have been known to mankind since the 9th century AD. Copper, silver and golden nanoparticles (NPs) were used to achieve a glim or to make colorshifting pottery. In 1847, Michael Faraday started investigating the properties of light and matter of nanoparticles. Faraday made slices of gold and shone a light on these. In order to produce gold thin enough to let light through, he used chemicals to make thin films. In the process of making these thin gold films he noticed that the water with which he washed the films would end up with a faint ruby colour. He shone a beam of light through the suspension and observed how the light formed a coneshaped figure in the water. Faraday suspected that gold particles in the suspension that were too small to detect were scattering the light to the side. Faradays’ studies would be the basis for modern nanoscience and nanotechnology [1].

The effect that was described by Faraday is caused by the difference in the physical prop-erties of metal NPs and those of metal in bulk. This difference is caused by specific electronic structures and a large reactive and exposed surface area. Because of these characteristics, NPs are interesting for applications in various research fields as solar energy, electronics and the food industry. Technology for synthesis of NPs has developed fast in the last decades. These developments have made it possible to modify and improve the particles and resulted in endless applications [2]. One of the characteristics of nano particles is their efficiency in scattering and absorbing light. Light of a certain wavelength turns nanoparticles into efficient nanosources of heat [3]. Under illumination at their plasmonic resonance wavelength, gold nanoparticles can absorb incident light and turn into heatsources which can be controlled by light remotely. This feature is used in thermal therapy; when a nanoparticle is excited by a laser beam, a part of the energy from the light will be converted to heat and will be transferred to the surrounding medium. The golden nanoparticles are targeted to tumour cells and when a laser produces non-ionizing electromagnetic radiation, the energy that is turned into heat by the nanoparticles and the cancer cell is killed. [4]. Another promising application is the use of nano particles to enhance light absorption in perovskite solar cells or to achieve high storage densities in optical data storage [5] [6]. The research field in which plasmonics and its applications are investigated is called thermoplasmonics.

The combination of the rapid development in nanotechnology and promising practical applications of NP’s has increased research’ efforts. The process of temperature dissipation is however not yet understood. The research group of Elizabeth von Hauff at the VU has started measuring the temperature using Surface Enhanced Raman Spectroscopy (SERS). In this research, the heat dissipation of gold NPs is modelled and compared with experimental results.

The question that is discussed in this research is: Can the heat dissipation of plasmonic gold nanoparticles be modelled? In order to answer this question, an adaptation of an existing MATLAB model was made in order to compare the outcome with experimental data. In this thesis, first the process of heating is explained. Then, a brief overview of SERS is given. Thereafter, results of the model are presented. This research concludes by comparing the results from the calculations with measurements and setting out directions for further research.

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3 Plasmonics and heat

In this chapter, first the phenomenon of plasmonic particles is explained followed by an elaboration on the heating of plasmonic particles. We conclude with the theory on calculating the heating of the particle.

3.1 Nanoplasmonics

Nanoparticles are particles consisting of a few hundred atoms. Their size varies from 1 to 100 nanometers [2]. Because of their size, their chemical and physical properties differ from bigger particles. This is caused by the surface to volume ratio. NPs exist in various shapes such as spheres, shells and rods. Both shape and structure of the NP’s characterize their reactivity. The optical properties of the particle depend on the size of the particle.

These optical properties are especially interesting in nanoparticles from noble metals. Noble metal NPs can absorb light in the near infrared region (650-900 nm). Gold, silver and copper all show these effects.Gold is used often because of the relative high stability. Furthermore, the synthesis of gold NPs is well practised. Because gold NPs have many free electrons, they are good absorbers and scatterers. Like other physical properties as electrical conductivity, this is caused by the presence of free conduction electrons. The free electrons move in a background of fixed positive ions. This ensures overall neutrality and plasma formation.

The free electrons of the particle can be excited by the electric component of light to have collective oscillations. To describe these electron collective oscillations , the term plasmon was introduced. A plasmon is a quantum quasi-particle representing the elementary excitation of the charge density oscillations in a plasma . When light shines on a molecule, the plasmon causes strongly enhanced surface plasmon resonance . This results in a strengthened electromagnetic field and a great enhancement of the absorption of light of the metal. This makes noble metals efficient scatterers and absorbers of visible light. [4]

3.2 Plasmons

Two important types of plasmons are localized surface plasmon-polaritons (LSPP) and prop-agating surface plasmon-polaritons (PSPP) [7].Both are described in this section.

PSPP

PSPPs are electromagnetic excitations that exist on the surface of a metal. The polari-tons propagate at the interface, between a dielectric and a conductor. The excitation is two-dimensional and the electromagnetic fields decay exponentially with distance from the surface. The waves lead to an enhancement of the electromagnetic field at the interface [8]. These electromagnetic surface waves are a result of the coupling of the electromagnetic fields to oscillations of the conductor’s electron plasma.The PSPP modes are TM or p-polarized electromagnetic waves. This nature of polarization is conserved at planar interfaces and therefore only TM waves can excite PSPP modes. PSPPs are confined to the perpendicular direction [9]. In figure 1 the p-polarized wave propagate in the x1-direction. The magnetic

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Figure 1: Perpendicular electromagnetic surface waves.

LSPP

On non planar surfaces such as metallic particles, the approximation used to describe the PSPP is no longer valid. For cases in which the particle is much smaller than the wavelength of the incident light (d << λ), localized surface plasmon-polaritons (LSPPs) are considered. [10] The LSPPs are the non-propagating excitations of the conduction electrons of metallic nanostructures coupled to the electromagnetic field. These modes arise naturally from the scattering of small, sub-wavelenght conductive nanoparticles in an oscillating electromagnetic field. This effect leads to field amplification both inside and in the near-field zone outside the particle. This resoncance is called localized surface plasmon resonance. If the size of the particle is much smaller than the wavelength of the light the quasi-static approximation can be used to understand the electromagnetic field [8].

Figure 2: The localized surface plasmon resonance.

The LSP modes of the nanoparticle will be excited by an incident wave with the appropriate polarization and frequency. At the LSP frequency, the coupling to LSP modes will result in a resonant optical response.

3.3 Heating

The surface plasmon resonance can trigger different processes. One of them is heating. In this paragraph, this process is explained. The incident light that hits the photon is either absorbed

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or scattered. When the light is absorbed, the energy is converted to heat. The conversion of the absorbed light to heat happens through three processes. These are illustrated in figure 3. The first two processes allow for the formation of a hot metallic lattice in the metal NPs and the last one allows for cooling off, i.e. heat dissipation to the surrounding environment.

Figure 3: The photothermal light to heat conversion by plasmonic nanostructures. In step one, the metal nanoparticles are excited by the incident light photons. This results in particle oscillation and charge separation. Second, the light is conversed to heat through electron-electron relaxation an electron-phonon relaxation processes. This results in the formation of hot metallic lattice. Third, heat dissipation is caused by electron-phonon coupling and phonon-phonon relaxation which cools the metal structure.[4]

3.4 Absorption and scattering efficiencies

A photon from an incident beam will either be absorbed or scattered by the particle. In order to determine the process of heat in gold NPs, it has to be known how much photons are absorbed by the plasmonic particle. Prashant et al. used Mie theory and disrete dipole approximation method to calculate absorption and scattering efficiencies for gold nanospheres [11]. Formulas 1 to 5 show the chance that the particle is either absorbed or scattered. The resonance wavelength, the extinction cross section and the scattering to absorption ratio depend on the size of the nanoparticle. The optical cross section differs per shape. Nanoshells for example have bigger optical cross sections. They calculated the cross sections for three classes of nanoparticles.

For gold nanospheres, the calculated absorption Qabs and scattering efficiency Qsca

is calculated on basis of Mie theory for homogenous spheres.

Qext= 2 x2 ∞ X n=1 (2n + 1)[a2n+ b 2 n] (1) Qext= 2 x2 ∞ X n=1 (2n + 1)Re[an+ bn] (2)

Qabs= Qext− Qsca (3)

an=

mψn(mx)ψn0(x) − ψn(x)y0n(mx)

mψn(mx)ξn0(x) − mξn(x)ψ0n(mx)

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bn=

ψn(mx)ψ0n(x) − mψn(x)y0n(mx)

mψn(mx)ξ0n(x) − mξn(x)ψn0(mx)

(5) In these efficiencies is a particle-specific optimum. For different sizes, the calculated extincition wavelength is shown in 1.

Table 1: Calculated Extinction Wavelength maxima of gold nanospheres

Size λM axnm

20 nm 521 40 nm 528 80 nm 549

The dimensionless efficiencies Qabs, Qef f and Qsca can be converted to the

cor-responding cross sections Cabs etc. by multiplication with the cross sectional area of the

nanoparticle. The optical cross sections of the gold nanoparticles are typically 4-5 orders of magnitude higher compared to those of conventionally used dyes. With the cross section, the amount of particles that is abosrbed can be calculated. With this information, the first step into calculating the temperature can be taken.

3.4.1 Maximum temperature

When doing calculations on the heat of a plasmonic particle, it helps to have an idea on the maximum temperature a particle can reach before it melts. This is what is described in this section.

When executing measurments on gold particles, the nanoparticles are generally in contact with a solid substrate or dissolved in a suspension. The effect of the interface between the NP and it surroundings has to be taken into account. Lee et al. modelled the thermody-namic processes at the surface using the CALPHAD (CALculation PHase Diagram) method [12]. Though the study focused on the melting of nanoparticles, it gives an idea of the max-ima. Gold nanoparticles on three different substrates were investigated. The study shows that through the substrates, the particles have a different melting temperature.

In figure 4 we see that the bulk melting temperature of 1064 degrees Celsius quickly drops as the particle radius gets smaller. At the radius of 1,5 nanometer the melting point will be 500 degrees Celsius.

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Figure 4: The melting temperature of gold NPs of different sizes [13].

The heat capacity of gold depends on the temperature of the material [14]. At room temperature, the heat capacity is 25.31, at its melting temperature it is 26.40. For this estimate we therefore use 26 J K−1mol−1. The weight of a particle of 5 nm will be about 0, 3 · 10−27. Using the heat capacity we can estimate the energy needed to heat up the particle one degree on 1, 5 · 10−24J .

The energy needed to heat a particle with radius 10 nm is 1, 1 · 1017_{J . In reality,}

the scattering efficiency will be lower so more energy is needed.

3.5 Heat transfer to the surrounding medium

The transfer of heat from the particle to the surroundings occurs at the surface of the particle. At macroscopic scale, one can assume that temperature experiences no discontinuity at the surface of a particle. On nanoscale however, the temperature and heat flux may no longer be continuous across the interface of two phases. This phenomenon was first described by Kapitza. Temperature at the interface changes proportional to the normal component of the heat flux through the interface, as the following formula states:

Tout− Tin= −RK· qinterf ace (6)

With RK, Kapitza or the thermal contact resistance and qinterf ace the heat flux

through the surface of the particle [15]. When calculating the heat dissipation from a plas-monic particle, a certain impedance at the surface has to be considered.

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4 Calculating the temperature of gold plasmons

In order to understand the heat dissipation of plasmonics, the first step of the project is to calculate the temperature of the particle. The process in the most simple way: what amount of energy enters the system, what is the amount that exists in the system. The difference is the increase in temperature. The energy going in, depends on the power of the laser beam and on how much the particle adsorbs. In order to know the amount of energy that is passed on to the surrounding, we need to know what is lost. Energy is lost transferring from one surface to another.

If we dive in deeper we want to know what happens when particles have different sizes: they can absorb more photons, but they have a smaller surface compared two volume, do they become hotter themselves? Or more efficient in sending heat into the surroundings? Secondly, what happens in time? When the laser is turned on, how long does it take for a equilibrium to be reached. And when it is turned off, how long does it take for the system to return to thermal equilibrium.

From the literature the absorption and scattering efficiencies, the extinction wave-length and the melting temperature of particles can be found. In order to calculate the temperature we need the cross section of the particle as well. To measure the dissipation of heat, the heat equation can be used. This is described in the following paragraphs.

4.1 Cross section of the nanoparticle

For calculating the temperature of the nanoparticle, the amount of photons incidenting has to be known. This depends on the size of the nanoparticle. Considering a single nanoparticle that is shone upon with monochromatic light, like a laser this is as follows: the power of the incident beam P , is proportional to the number of incident photons per unit time. The power density SIncin W m−2is expressed by:

SInc=

2PInc

πω2 0

(7) The intensity or power P [W ] that hits the nanoparticle depends on the cross section σ, the effective area of a homogeneous incoming beam from which the nanoparticle will absorb every photon, times the power density [7]:

P − σSInc (8)

Keeping in mind, that seeing the cross section simply as the surface won’t give a precise calculation. In order to get it more precise, we need to define a cross section for every vibrational mode of the particle. For the golden nanospheres, the cross sections were calculated by Mie. [11]

4.2 Heat equation

In order to calculate the heat in the surroundings of the particle, the heat flux through the particle has to be calculated. We use the heat equation to calculate it further. Since we are working on a round nanoparticle, we use the sphericle heat equation.

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1 r2 δ δr kr2δT δr + 1 r2_sin2_θ δ δφ kδT δφ + 1 r2_sinθ δ δθ ksinθδT δθ + qV = ρcp δT δt (9)

Since the heat dissipation is symmetrical, we only use the radial part:

1 r2 δ δr kr2δT δr + qV = ρcp δT δt (10)

If we look at the steady state we can take out the time dependence.

1 r2 δ δr kr2δT δr + qV = 0 (11)

We can then solve it into Fouriers law:

qr = −kAδT

δr (12)

With A the volume of the sphere: qr 4πr2 = −kδT (13) qr 4π Z ri r δr r2 = −k Z Ti T δT (14) After inserting qr= 4πk(Ti− T0) 1 ri − 1 r0 (15) T (r) = Ti− (Ti− To) 1 ri − 1 ro 1 ri −1 r (16)

With qV the rate at which energy is generated per unit volume of the medium

[W m−3] and k the conductivity of the material [W m−1K−1].

In order to calculate the heat flux, the power of the laser, the absorption and the conversion are taken into account.

4.3 Reaching equilibrium

When adding energy to the system, the system isn’t suspected to heat up infinitely, at one point it should reach an equilibrium. Locally, this will happen quite fast, farther away from the particle, the equilibrium will be reached slower depending on the heat capacity in the surroundings. The energy dissipating from the system Qextis expressed as:

Qext= hS(T − TO) (17)

Here h is a heat transfer coefficient and S is the cross sectional area perpendicular to conduction. Qextwill be zero when T becomes equal to TO [16].

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5 Modelling the temperature

In the previous chapter, some theory and calculations on heating were described. In this chapter, the proces of modelling this is described.

5.1 The existing MATLAB model

In a previous project a MATLAB finite differences model that shows a heat map of a single or a group of nanoparticles was build. The in-built ’heat-map’ color scheme of MATLAB is used to show the increase in temperature. The matrix starts out all black, which represents the starting temperature. If a laser excites the particles, they get warmer and thus their colour changes from black, via red to white that represents an increase of more than 100. The more time elapses, the more heat will appear on the map. In figure 5 a still from the animation of the heat dissipation can be seen.

Figure 5: Still from the animation made by the MATLAB model. The scale shows how the colours relate to the increase times 100.

The model gives a nice impression of the spreading of the heat. The model was used as a start for this project. The recommendations for improvement of the model were stated in the previous project:

1. Physical values have to be added to the model

2. Different starting distributions have different results over time

3. The particle in the model is one block, it could be interesting to investigate bigger particles and particles made of different sizes.

After the literature research we can add the following points:

1. With a high input energy that is transferred to warmth, the temperature can not exceed the melting temperature of gold.

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2. The warmth shouldn’t spread infinitely, at some point the model should reach an equi-librium.

3. The heat dissipation at the surface of the particle is different than through the medium. 4. At the surface of the particle, the spreading depends on the size of the particle, the

surrounding matrix and other surface effects.

5.2 Adapting the model

A model can be either steady state or time dependent. Thereby it can resemble one particle or multiple particles. It is a time dependent model that can be used on both one and on multiple particles. In order to compare the results of the model to temperature measurements, it has to show temperature and temperature distribution. Therefore the model with one particle was used.

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6 Experimental results from literature

In order to check the data obtained from the model, experimental results are needed to verify

the model. Service Enhanced Raman Spectroscopy (SERS) is a promising technique for

measuring the temperature of plasmonics. However, SERS measurements with usable results were not found. Therefore other data to compare the results with were searched for. In this section, different measuring techniques are described. First the Pozzi research to SERS is briefly elaborated on, then the Richardson study on plasmonics in water is described.

6.1 Measuring the temperature

In order to measure the temperature of nanoscale systems, different techniques are being tried. The problem is that measuring instruments will influence the temperature and that on nanoscale, the temperature can vary locally because of hotspots. With methods as measuring with optical tweezers [17], using white light scattering spectroscopy [18] and Surface Enhanced Raman Spectroscopy (SERS) [19] it is tried to solve these difficulties.

SERS

SERS is a promising method for measuring the temperature of nanoscale systems. It uses the amplification of electromagnetic fields generated by the excitation of the localized surface plasmons to deduct the temperature. This way, it could be a sensitive detection method for low concentration analytes [20]. In Appendix A, an overview of the working on SERS can be found. Since the results were not precise, we couldn’t use it to compare our model in. We chose an other experiment to do so. In the future, after development of more precise SERS results, comparing the model to those results will be interesting.

Figure 6: The results of the Pozzi experiment: from the left figure, the broad distribution in the ratio of anti-Stokes and Stokes signal intensities can be seen.

Nanoparticles in a water droplet

Richardson et al. conducted an experiment in which the light to heat conversion of gold NPs in a water droplet was investigated. The NPs in the droplet were excited by a laser and the

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temperature change in the droplet is measured. [16]

The outcome of their experiment is plotted in figure 7.

Figure 7: The result of the Richardson experiment.

In the next chapter we compare this result to the result of the previously discussed MATLAB model.

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7 Modelling results

In this chapter the results from the model are discussed. In order to do so, the model was run with different variables. First the temperature increase of the particle as a result of heating is modelled. Than the heat pattern in the surroundings of the particle is calculated by the model. The results are plotted and compared to the heat pattern calculated by the heat equation. Lastly, the heating and cooling of the particle in the model are combined in one graph and compared to the result of experimental data.

7.1 Temperature heated particle

In section 3.4.1. the melting temperatures of gold NPs are plotted in figure 4. This is the maximum temperature the particle should reach. With the model, we plotted the increase of energy of a center particle which is consistently heated by a 1 W laser. The result can be seen in figure 8. We see in the graph that it approaches an equilibrium of 30J which corresponds to a temperature increase of a temperature way above the melting temperature of a gold particle of this size, which is shown in figure 4.

Figure 8: We plot the energy increase of the center particle if we keep heating it by a laser, in this case a laser of 1J/s/nm2 _{for a particle of 10 nm radius}

We estimated the energy needed to heat up a NP with a radius of 5 nm to be 1, 5 · 10−24J . When we compare this with figure 8 we see that the particle would melt within 0,1 seconds of the laser shining on the particle. We should therefore either use a laser with a lower intensity or use a laser that pulsates in order to prevent the particle from melting.

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7.2 Equilibrium state compared to heat equation

In figure 9 we compare the the theoretical heat equation as derived in section 4.2 with the equilibrium state as given by the MATLAB model.

(a) (b)

Figure 9: (a) A plot of the Heat equation explained in section 4.2 (b) The equilibrium state temperature for the system with a constant temperature of the heated gold particle

We see comparable behaviour for both the mathematical model and the MATLAB model, but it is also immediately clear that the mathematical description gives a much steeper energy decrease than the MATLAB model. One clear distinction between the two models is that the MATLAB model assumes a square system, which also corresponds to realistic measurement set-ups, while the mathematical model is for a circular system.

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7.3 Timed pulses

The particle in the MATLAB model is heated up by a pulse and than cools off again. This clearly mimics experiment of Richardson et al. [16], figure 10a, in which a water droplet was heated for a pulsed time.

(a) (b)

Figure 10: (a) The result by Richardson et al [16] (b) The result of the model by using a timed pulse which starts at 4s and ends at 12s

The simulated result follows the trend of the experimental results of Richardson et al. However, the modelled system does not return to 0 temperature increase as quickly as the experimental result. This is the same behaviour as we found in the previous section where we compared the heat equation to the energy distribution.

As is explained in section 3.5, on the surface of the particle an impedance could play a role in the dissipation of the heat. In the model, the amount of energy passed to the surroundings is therefore 1 minus the edge impedance. This is further elaborated on in Appendix C.

By setting the edge impedance to 1, the particle should not pass on heat. However, if we set it to 1 we get the result shown in figure 11. So the energy is definitely lost from the particle to it’s surrounding substrate. Furthermore this behaviour more closely resembles the experimental behaviour that was found by Richardson et al.

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8 Discussion

The goal of this thesis, is to gain more insight on the heat dissipation of golden NPs. With a MATLAB finite differences model, this process is modelled and compared to calculated and experimental results.

First, a single particle was heated and the energy increase was plotted. This increase results in an unrealistic temperature since it is above the melting temperature of the gold NP. In the plot that shows this, the absorption efficiency was not taken into account. This should lead to a smaller amount of energy absorbed and therefore a lower temperature. The difference will not be big enough to make a difference; the particle would still melt. Since the system will reach an equilibrium at one point, the temperature will not rise infinitely and will be constant when run for a longer period.

In the figure in which the temperature dissipation in the surroundings is plotted, figure 10 b, some irregularities can be spotted. This can either be caused by the fact that the model did not take enough time steps to get a good result. An other option is that the model is simulating a sphere in a square. The third option is that the impedance that was used is causing trouble.

In figure 10 the result of the model is compared to the result of an experiment. We are comparing the shape here, since we plotted the energy absorbed by the particle. We see that the shapes are similar though in the experiment, the particle cools of to its first temperature faster. Trying different settings on the model, gives the idea that setting the impedance on 1, gives the best result. This seems not right, since the model implies that in this case no heat will be transferred to the surroundings of the particle.

In figure 11 the impedance is set to 1. The particle should not pass on any heat. Since the heating and cooling of is more abrupt than when the impedance is lower, the opposite appears to happen. This implies that this part of the model is not working correctly. Furthermore, a low impedance at the edges appears to be more similar with the experimental result. This implies that the edge impedance would have a small to negligible role in the dissipation.

When a SERS experiment has been conducted, the influence of the electric field caused by the plasmonic effect on the temperature could be investigated.

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9 Conclusion and outlook

The objective of this thesis was to build further on a model that would predict the process of heat dissipation of golden nanoparticles.This was done by adapting a MATLAB model made during a previous project and comparing it to experimental results. The goal is to gain a better understanding of the process of heat dissipation.

In the existing model, physical values from literature about conducted experiments could be put in. The results of this model would then be compared to measured values from the literature the physical values came from. The plan was to use results from SERS measurements. After thorough literature research to get a good understanding of plasmonics and SERS the first result stumbled upon was unexpected. This was that there were no articles describing successful SERS.Therefore, in this thesis the outcome of the model is compared to the outcome of research done by Richardson. The model was able to show the same result as the result from the experiment.

The results show that the modelled results resemble both the result calculated with the heat equation and the experimental results from literature. From trying different settings in the model, it can be concluded that for the spreading of the heat, the heat capacity is of a far greater influence than the edge impedance.

Questions for follow up research would be:

1. Could the model predict the pattern of heat dissipation for particles of different shapes and sizes?

2. Developments in the field of SERS have taken place: could we use the model to check new research?

3. A model that displays the electrical field is made in a different study, can the two models be combined and show the influence of both effects?

By adding more components and physical values to the model and comparing this with experimental results, more theories about the heat dissipation of NPs can be obtained. With a better understanding, the use and application of the NPs can be boosted.

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[6] Masud Mansuripur, Armis R Zakharian, A Lesuffleur, Sang-Hyun Oh, RJ Jones, NC Lindquist, Hyungsoon Im, A Kobyakov, and JV Moloney. Plasmonic nano-structures for optical data storage. Optics express, 17(16):14001–14014, 2009.

[7] Eric Le Ru and Pablo Etchegoin. Principles of Surface-Enhanced Raman Spectroscopy: and related plasmonic effects. Elsevier, 2008.

[8] Anatoly V Zayats, Igor I Smolyaninov, and Alexei A Maradudin. Nano-optics of surface plasmon polaritons. Physics reports, 408(3-4):131–314, 2005.

[9] Stefan Alexander Maier. Plasmonics: fundamentals and applications. Springer Science & Business Media, 2007.

[10] Anatoly V Zayats and Igor I Smolyaninov. Near-field photonics: surface plasmon po-laritons and localized surface plasmons. Journal of Optics A: Pure and Applied Optics, 5(4):S16, 2003.

[11] Prashant K Jain, Kyeong Seok Lee, Ivan H El-Sayed, and Mostafa A El-Sayed. Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine. The journal of physical chemistry B, 110(14):7238–7248, 2006.

[12] Joonho Lee, Toshihiro Tanaka, Junggoo Lee, and Hirotaro Mori. Effect of substrates on the melting temperature of gold nanoparticles. Calphad, 31(1):105–111, 2007.

[13] G¨unter Schmid and Benedetto Corain. Nanoparticulated gold: syntheses, structures, electronics, and reactivities. European Journal of Inorganic Chemistry, 2003(17):3081– 3098, 2003.

[14] Yoichi Takahashi and Hidetoshi Akiyama. Heat capacity of gold from 80 to 1000 k. Thermochimica acta, 109(1):105–109, 1986.

[15] WJ Minkowycz, Ephraim M Sparrow, and John P Abraham. Nanoparticle heat transfer and fluid flow. CRC press, 2016.

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[16] Hugh H Richardson, Michael T Carlson, Peter J Tandler, Pedro Hernandez, and Alexan-der O Govorov. Experimental and theoretical studies of light-to-heat conversion and collective heating effects in metal nanoparticle solutions. Nano letters, 9(3):1139–1146, 2009.

[17] J Millen, T Deesuwan, P Barker, and Janet Anders. Nanoscale temperature measure-ments using non-equilibrium brownian dynamics of a levitated nanosphere. Nature nan-otechnology, 9(6):425, 2014.

[18] Tobias Dominik Gokus. Time-Resolved Photoluminescence and Elastic White Light Scat-tering Studies of Individual Carbon Nanotubes and Optical Characterization of Oxygen Plasma Treated Graphene. PhD thesis, Citeseer, 2011.

[19] Eric A Pozzi, Alyssa B Zrimsek, Clotilde M Lethiec, George C Schatz, Mark C Hersam, and Richard P Van Duyne. Evaluating single-molecule stokes and anti-stokes sers for nanoscale thermometry. The Journal of Physical Chemistry C, 119(36):21116–21124, 2015.

[20] Bhavya Sharma, Renee R Frontiera, Anne-Isabelle Henry, Emilie Ringe, and Richard P Van Duyne. Sers: Materials, applications, and the future. Materials today, 15(1-2):16–25, 2012.

[21] Xu Xie and David G Cahill. Thermometry of plasmonic nanostructures by anti-stokes electronic raman scattering. Applied Physics Letters, 109(18):183104, 2016.

[22] Yukie Yokota, Kosei Ueno, and Hiroaki Misawa. Highly controlled surface-enhanced raman scattering chips using nanoengineered gold blocks. Small, 7(2):252–258, 2011.

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11 Appendix

A

Surface Enhanced Raman Spectroscopy

Measuring the temperature of nanoparticles was tried in various ways. Because of the scale on which the temperature has to be measured, gangbare measuring equipment cannot be used, since anything added to the system will distort the temperature resulting in a less precise result. Optical tweezers, geographical mapping, in body measurments are ways that seem promising; a breakthrough hasn’t been achieved. A promising method is Surface Enhanced Raman Spectroscopy (SERS) with which we can measure the signal of the sample close to the particle. We start with the principles of Raman Spectroscopy and then describe the difference with SERS. At the end of the chapter it is explained how the temperature can be measured with SERS.[7] In order to understand to process of SERS, we start by describing Raman Spectroscopy and from there investigate SERS and experiments that have been conducted so far.

A.1 Raman Spectroscopy

In Raman Spectroscopy a laser beam is used to illuminate a sample. The sample is made by attaching a probe, a molecular species, to a metallic structure, the substrate. As a photon from the light beam hits a molecule, it can excite it to a higher energy level. The photon energy E is hereby absorbed by the molecule. The new energy level EB of the molecule will

be higher than the old energy level EA:

E = Eb− Ea (18)

The energy E[J ] of the incoming photon can be expressed with ω its angular frequency, ν its frequency, λ its wavelenght and T the temperature in Kelvin:

E = hν = ~ω = hc

λ = kbT (19)

The opposite of absorption is emission: the molecule will emit a photon while lowering from higher level of energy Eb to Ea. This can only happen if the molecule isn’t in ground state

but an excited state. The simplest and most common form is the excitation of a molecule from its groundstate S0 to S1, the first excited state.

The interaction between molecules and light is primarily determined by the energy levels of the degrees of freedom of the molecule. These energy levels are visualized by using a Jablonski diagram.

If the molecule absorbs a photon but emits a photon at the same time, this process is called scattering. When both the incident and scattered photons have the same energy, the molecule will have the same energy as before. This elastic scattering is called Rayleigh scattering or in nanoparticles: Mie scattering. If the incident and scattered photons don’t have the same energy, the energy of the molecule changes. This is inelastic scattering, called Raman scattering.

The absorbed and scattering occur at the same time. However, for molecule to emit a photon, it has to be in a higher energy state. If we think of Raman scattering in two steps: first a photon will be absorbed which will excite the electron to a state that might not exist.

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This state is called a virtual state, a mathematical construction of perturbation theory. If this mode does exist, resonant scattering will occur.

A.2 Raman spectrum

When Raman scattering occurs, (inelastic scattering) the emitted photon has either a higher or a lower energy than the incident photon. If the energy of the emitted photon is lower, the electron will be excited to a higher energy level, after the scattering has taken place. This is called a Stokes process. If it is the other way around, it is called anti-Stokes. The difference in energy is called the Raman shift. And is defined by the following equation:

∆ER= EL− ES (20)

Therefore, for a Stokes process ∆ERwill be positive, for a anti-Stokes process it will

be negative. Raman shifts are expressed in wave-numbers and denoted ∆νRIn case of Stokes

scattering, the energy that transfers from the photon to the molecule, results in a measurable red-shift. In case of Anti-stokes scattering, the energy decreases which results in a blue-shift. This can be seen in the Raman spectrum, we see the laser in the middle,

The Raman spectrum corresponds to the wavelength or energy dependence of the Raman scattered intensity at a given incident wavelength. The peaks in the Raman spectrum correspond to the vibrational modes of the molecule.

A.3 SERS

SERS, is Raman Spectroscopy with an amplified signal. The electromagnetic interaction of the laser light with metal causes the laser field to be amplified which causes plasmon resonance. The Raman signals of the molecules are enhances by the plasmon resonance. An obvious property that influences this enhancement is the surface of the particle, the bigger the surface, the larger the SERS signal.

A.4 Enhancement factor

The quantity with which the Raman signal is enhanced compared to normal conditions is called the enhancement factor (EF). The average SERS EF is the EF averaged over all the possible positions on the metallic surface. It should be easy to achieve average SERS EFs between 105and 106, the good substrates will have an EF as large as 107− 108

.

The Raman intensity of a molecule depends directly on the laser power density and the Raman cross section of the molecule. In SERS, this should be the same except that it will be affected by the enhancement factor. The Enhancement factors are split in two main types: electromagnetic and chemical enhancement. Electromagnetic enhancement is caused by the coupling of the incident and Raman electromagnetic fields with the SERS substrate, both for the incident and the re-emitted field. The chemical enhancement is believed to be much smaller, assuming that it even exists (not everyone agrees on that). Nevertheless, it is not necessary to know how EF works in order to try to calculate it.

”The local field enhancements depend strongly on the exact position of the molecule on the surface’. Since the distribution of molecules on a SERS substrate is not homogeneous.

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The SERS EFs are highly non-uniform. Points where a large enhancement takes place, as so-called hot-spot, can lay within tens of nanometers of points with little enhancement. When performing Single Molecule SERS, these hotspots can be used to get clear results, however, in other cases this is not as handy.

A.5 Measuring Temperature with SERS

The ratio between Stokes scattered photons and anti-Stokes scattered photons depends on the vibrational density of states Nn and Nm as is given by the Boltzmann distribution:

Nn Nm = gn gm exp −(En− Em) kBT (21) Here gnand gmare the degeneracy of the levels, Enand Em are the energy levels,

kB is the Boltzmann constant, T is the absolute temperature. By knowing the occupancy of

the energy levels we can calculate the temperature. If we can measure the occupancy of the energy levels by recording the Stokes and anti-Stokes Raman signals, we can calculate the temperature. The ratio of Stokes S to anti-Stokes aS signals, ρ is expressed as:

ρ =IaS IS − A τ σ 0 SIL hνL + exp(−hνm kbT ) (22) In this formula, vm is the frequency of the Raman band with vibrational excited

state lifetime τ , Raman cross section σS0, local temperature T and asymmetry factor A which

is expressed as: A =ηaS ηS σ0aS σ0 S ELEaS ELES 2 (23) This asymmetry factor is the difference between the Raman intensities and the population intensities. With η the wavelength-dependent detection efficiency of the detector, E the local electric field strength and sigma0the Raman cross section. Which is described in the Pozzi experiment as:

σm0 = σ0,mνL(νL± νm)3 (24)

In the Pozzi experiment, they tried to use these formules to use Raman spectroscopy as thermometric method. They tried to measure silver nanoparticle SERS substrates using rhodamine 6G as Raman scatterer. [19]

A.6 Research

A.6.1 Pozzi

Pozzi used glass slides coated with AgNP that were treated with R6G, both a deuterated and a non-deuterated variant. The confocal Raman microscope had an excitation wavelength of 532 nm and had 11 different laser irradiance powers between 2 and 178 mW. The correlation, ρ varied over 1.5 orders of magnitude. The average correlation between integrated Stokes and anti-Stokes intensities was 0.42, not quite the ideal value of 1 which would indicate perfect

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correlation. This is consistent with the imperfect correlation of the integrated anti-Stokes and Stokes intensities.[19] The authors expect that the Stokes and Anti-Stokes molecular cross sections remain relatively constant. The factors remaining that can cause the variance in ρ are relative intensity fluctuations due to changes in excited state geometry, variations in the EF between anti-Stokes and Stokes transitions, local heating and vibrational pumping into the v = 1 state. Since the last one vibrational pumping, was found to be conducted at significantly higher power densities, this one was found to be negligible. From literature about another research the The three most important factors. The fluctuations in excited state geometry: large intensity fluctuations because of the overlap between wave functions of particular vibrational states which causes the anti-Stokes and Stokes SMSERS of the same band to vary unevenly. EF wavelength dependence: EF appears to be a significant contribu-tor to ρ. As shown in equation ?? above. And wavelength dependent. Therefore, Stokes and anti-Stokes light might be enhanced differently. In the paper it was concluded that the EFs are highly dependent of the wavelength of the scattered photon. This results in the conclusion that rho can be heavily influenced by EF differences in the near field. Local heating: The two assumptions made are: 1) molecules at EM hot spots will give rise to the highest SERS signals and 2) at stronger hot spots, more local heating is expected. This means that greater heating correlates with higher SERS intensity when local heating occurs. When tested, pos-itive correlation suggest that local heating is happening. The local heating could cause an increase in the diffusion of mobile molecules from locations with a high temperature. This way, the temperature is systematically lowered though the real temperature doesn’t change. Since the relation is positive, the authors suspect that local heating is observed though the EF wavelength dependence withholds them from accurate temperature measurements. From simulating the EF they learned that the EF probably varies between anti-Stokes and Stokes transitions. Also, the wavelength dependence of enhancement depends on the location of the molecule. Outlook: Because of the mentioned variations, determining ρ is not straightfor-ward. Especially the EF variations, they can’t be overcome by measuring in a different way. According to Pozzi, once a EM field is characterized, an accurate local temperature can be calculated when corrections are made for the other factors affecting ρ.

A.6.2 Xie

Xie and Cahill performed a temperature measurement on golden nanodisks. They heated the individual nanodisks by a focused laser beam and measured their temperature from the anti-Stokes using a description based on electronic Raman scattering. [21]

A.6.3 Yokota

Yokota performed SERS on gold nanostructures of various sizes on glass substrates. The results were compared to a finite-difference time-domain simulation using EM theory. The aim of the research was to make a quantitative SERS chip that could be used in other research. The conditions that had to be met were 1) the SERS signals must be highly reproducable; 2) the SERS signals must maintain a constant value for every measurement on different spots and 3) measurements must be performed in aqueous solution or organic solvents with a water or oil-immersed objective lens and 4) gold should be used. [22]

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B

MATLAB code

1 c l e a r a l l 2 3 % Choose v i s u a l scheme 4 c o l o r m a p(h o t) 5 a x i s o f f 6 a x i s s q u a r e 7 8 % S e t v a l u e s 9 N = 1 0 0 ; % S i z e x S i z e o f t h e g r i d 10 N t = 1 0 0 ; % Number o f t i m e s t e p s 11 T t o t = 1 0 ; % T o t a l t i m e 12 L o s s = 0 . 0 1 ; % L o s s ( i n p e r c e n t a g e ) a t t h e b o r d e r s 13 K au = 3 1 4 ; % W/ (m∗K) % Heat C o n d u c t i v i t y o f g o l d ( n o t u s e d ) 14 r s p o t = 1 % S i z e o f t h e s p o t i n which p a r t i c l e s a r e h e a t e d 15 i n c r e a s e = 10/p i % I n c r e a s e w i t h i n t h e s p o t l i g h t

16 Edge imp = 0 . 1 ; % P e r c e n t a g e impedence on t h e e d g e s o f t h e

p a r t i c l e s 17 18 %% When a l l t h r e e f a l s e i t i s t h e o r i g i n a l model 19 p l o t h e a t = t r u e 20 21 c o n s t v a l u e = f a l s e 22 c o n s t v a l = 100 23 24 t i m e d p u l s e = f a l s e 25 s t a r t p u l s e = 1∗ T t o t /5 26 e n d p u l s e = 3∗ T t o t /5 27 28 29 30 % P a r t i c l e l o c a t i o n s / v a l u e s 31 % The b e g i n v a l u e o f t h e p a r t i c l e s must a l w a y s be > 0 32 S = z e r o s(N) ; 33 f o r i = 5 : 3 : (N−5) 34 f o r j = 5 : 3 : (N−5) 35 i f a b s(randn) <= 0 . 2 36 S ( i , j ) = 1 ; 37 end 38 end 39 end 40 41 42

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43 % C a l c u l a t e d v a l u e s 44 N mid = round(N/ 2 , 0 ) ; 45 dt = T t o t / N t ; 46 47 48 % Make c o n d u c t i o n m a t r i x C 49 C = o n e s (N) ; 50 [ i , j ] = f i n d( S ) ; % F i n d s a l l non−z e r o l o c a t i o n s i n S 51 f o r n = 1 :l e n g t h( i ) 52 C( i ( n ) , j ( n ) ) = (1− Edge imp ) ; 53 end 54 55 % Time e v o l u t i o n 56 f o r i = 1 : N t 57 t=i ∗ dt ; 58 So = S ; 59 f o r x = 1 :N 60 f o r y = 1 :N 61 i f ( x==1 | | y==1 | | x==N | | y==N) % So ’ i f b o r d e r ’ 62 i f x==1 63 S ( x , y ) = (1− L o s s ) ∗S ( x+1 , y ) ; 64 e l s e i f x==N 65 S ( x , y ) = (1− L o s s ) ∗S ( x −1 , y ) ; 66 e l s e i f y==1 67 S ( x , y ) = (1− L o s s ) ∗S ( x , y+1) ; 68 e l s e i f y==N 69 S ( x , y ) = (1− L o s s ) ∗S ( x , y−1) ; 70 e l s e 71 ’ n o t s u p p o s e d t o happen ’ 72 end 73 e l s e

74 c r o s s s o m = sum( [ C( x , y+1)∗ So ( x , y+1) ,C( x , y−1)∗ So ( x , y

−1) ,C( x+1 ,y ) ∗ So ( x+1 ,y ) ,C( x −1 ,y ) ∗ So ( x −1 ,y ) ] ) ;

75 c o r n s o m = sum( [ C( x+1 ,y+1)∗ So ( x+1 ,y+1) ,C( x+1 ,y−1)∗

So ( x+1 ,y−1) ,C( x −1 ,y+1)∗ So ( x −1 ,y+1) ,C( x −1 ,y−1)∗ So ( x −1 ,y−1) ] ) ;

76 n e w v a l = 0 . 1 5 ∗ c r o s s s o m +0.10∗ c o r n s o m + (1−C( x , y ) )

∗ So ( x , y ) ;

77 S ( x , y ) = n e w v a l ;

78

79 i f ( ( x−N mid ) ˆ2 + ( y−N mid ) ˆ 2 ) ˆ ( 0 . 5 ) <= r s p o t

80

81 i f c o n s t v a l u e

82 S ( x , y ) = c o n s t v a l ;

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84 e l s e i f t i m e d p u l s e 85 i f s t a r t p u l s e <t && t<e n d p u l s e 86 S ( x , y ) = S ( x , y ) + i n c r e a s e ; 87 % e l s e n o t h i n g happens and i t l o s e s e n e r g y a s any 88 % p l a c e 89 end 90 91 e l s e 92 S ( x , y ) = S ( x , y ) + i n c r e a s e ; 93 end 94 end 95 end 96 end 97 end 98 %image ( S ) % Show t h e s y s t e m a t t h i s t i m e s t e p 99 %p a u s e ( 0 . 0 5 ) % T h i s i s t h e t i m e i n s e c o n d s how l o n g e a c h image r e m a i n s 100 t i m e l i s t ( i ) = t ; 101 t e m p c e n t ( i ) = S ( N mid , N mid ) ; 102 end 103 104 %% 105 i f p l o t h e a t 106 p l o t( t i m e l i s t , t e m p c e n t ) 107 t i t l e(’ t i t e l ’) 108 x l a b e l(’ d i s t a n c e ’) 109 y l a b e l(’ Energy i n c r e a s e ( J ) ’) 110 end 111 112 %% 113 i f c o n s t v a l u e 114 p l o t( S ( N mid , : ) ) 115 t i t l e(’ t i t e l ’) 116 x l a b e l(’ d i s t a n c e ’) 117 y l a b e l(’ Energy i n c r e a s e ( J ) ’) 118 end 119 120 %% 121 i f t i m e d p u l s e 122 p l o t( t i m e l i s t , t e m p c e n t ) 123 t i t l e(’ t i t e l ’) 124 x l a b e l(’ d i s t a n c e ’) 125 y l a b e l(’ Energy i n c r e a s e ( J ) ’) 126 a x i s([ − i n f i n f −1 i n f ] )

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C

Model Implementation

This Appendix introduces the used Matlab code, it is based on the description, of the original model. The first section C.1 describes in what parts and mathematical descriptions the system is split while the second section C.2 explains the implementation within Matlab.

C.1 Physical model broken up

A graphic representation of the model can be found in figure 12. It is a system with every NP having an constant add in of energy, if they are within the spotlight for the timed pulse we only have this increase for the duration of the pulse. For every time step the energy will spread out from those regions with high temperature to those with a lower temperature. The speed at which this happens depends on the thermal conductivity, which for gold is 314 W

m ˙K.

At the sides of the model we implement a constant percentage based loss with the imaginative variable name ’Loss’.

Figure 12: A graphic presentation of our model as explained in section C.1

C.2 Code Implementation

The model keeps track of 2 two-dimensional arrays. The first keeps track of the diffusing energy while the second is a list of the locations of all nano-particles. The particles are placed using the inbuilt random number generator -randn and only placing an particle if the generated number is higher than a certain threshold (currently set at 0.2). It does not set any particle in the first 5 blocks from any side, and never puts them close together.

Instead of having a time dependant diffusion we have all energy diffuse between neighbours unless the diffusion is obstructed by an edge impedance. Each new value not on the out most border of the sample is generated by a set of simple steps. First we calculate the values of the ’cross neighbours’, horizontal and vertical (Sx±1,yandSx,y±1). Than we calculate

the diagonal neighbours (Sx±1,y±1), all the time checking if the boxes house particles or not,

saved in array C. The implementation is that all elements in C are 1 except those holding a particle, whose value is (1 − Edge imp). When taking the sum of the neighbours we weight them all so their sum equals exactly one. Currently we weigh the cross neighbours 0.15 and the cross neighbours 0.10 . These numbers could be varied, keeping in mind that their sum should always be equal to 1.00 .

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Sum neigh(i, j) = wcross X

cross

Ci,jSi,j+ wdiag

X diag Ci,jSi,j (25) with: wcross X cross 1 + wdiagX diag 1 ≡ 1 (26)

Si,jt+dt= Sum neigh(i, j) + (1 − Ci,j) ∗ Si,jt (27)

If {i,j} is within the spotlight : Si,jt+dt= S

t+dt

i,j + Increase ∗ (1 − Ci,j) (28)

Within this project the code was extended to visualise the energy of a cross section of the system (7.2). Furthermore we show the energy of the center particle while continuously heating (7.1) and for a timed pulse (7.3).

C.3 Macroscopic model

Richardson et al. build a macroscopic model instead of a nanoscopic model. [16] They do try to put the experiment to a one particle answer

The model that was build starts with the following energy balance equation: X

i

miCp,i

dT

dt = QI− Qext (29)

With mi the mass and Cp,ithe heat capacity of the system, T temperature en t time. QI is

the energy supplied to the system, it is given by:

QI= I(1 − 10−Aλ)η (30)

In this equation, I is the incident power, Aλis the absorbance of the nanoparticle

solution and η the efficiency. With Aλ= lopt· C · with Iopt, C and are the optical path,

molar concentration and molar extinction coefficient. For Aλthe value 0.0217 found in the

Modelling heat dissipation from plasmonic gold nanoparticles