Heat generation with plasmonic nanoparticles

Heat generation with plasmonic nanoparticles

arXiv:1703.00422v1 [math.AP] 1 Mar 2017

Habib Ammari∗

Francisco Romero†

Matias Ruiz†

Abstract In this paper we use layer potentials and asymptotic analysis techniques to analyze the heat generation due to nanoparticles when illuminated at their plasmonic resonance. We consider arbitrary-shaped particles and both single and multiple particles. For close-totouching nanoparticles, we show that the temperature field deviates significantly from the one generated by a single nanoparticle. The results of this paper open a door for solving the challenging problems of detecting plasmonic nanoparticles in biological media and monitoring temperature elevation in tissue generated by nanoparticle heating.

Mathematics Subject Classification (MSC2000): 35R30, 35C20. Key words: plasmonic nanoparticle, plasmonic resonance, heat generation, Neumann-Poincar´e operator.

1

Introduction

Our aim in this paper is to provide a mathematical and numerical framework for analyzing photothermal effects using plasmonic nanoparticles. A remarkable feature of plasmonic nanoparticles is that they exhibit quasi-static optical resonances, called plasmonic resonances. At or near these resonant frequencies, strong enhancement of scattering and absorption occurs [5, 7, 27]. The plasmonic resonances are related to the spectra of the non-self adjoint Neumann-Poincar´e type operators associated with the particle shapes [5, 7, 8, 9, 15, 21]. Plasmonic nanoparticles efficiently generate heat in the presence of electromagnetic radiation. Their biocompatibility makes them suitable for use in nanotherapy [10]. Nanotherapy relies on a simple mechanism. First nanoparticles become attached to tumor cells using selective biomolecular linkers. Then heat generated by optically-simulated plasmonic nanoparticles destroys the tumor cells [14]. In this nanomedical application, the temperature increase is the most important parameter [23, 26]. It depends on a highly nontrivial way on the shape, the number, and organization of the nanoparticles. Moreover, it is challenging to measure it at the surface of the nanoparticles [14]. In this paper, we derive an asymptotic formula for the temperature at the surface of plasmonic nanoparticles of arbitrary shape. Our formula holds for clusters of simply connected nanoparticles. It allows to estimate the collective response of plasmonic nanoparticles. ∗ Department of Mathematics, ETH Z¨ urich, R¨ amistrasse 101, CH-8092 Z¨ urich, Switzerland ([email protected], [email protected]). † Department of Mathematics and Applications, Ecole Normale Sup´erieure, 45 Rue d’Ulm, 75005 Paris, France ([email protected]).

1

The paper is organized as follows. In section 2 we describe the mathematical setting for the physical phenomena we are modeling. To this end, we use the Helmholtz equation to model the propagation of light which we couple to the heat equation. Later on, we present our main results in this paper which consist on original asymptotic formulas for the inner field and the temperature on the boundaries of the nanoparticles. In section 4 we prove Theorems 2.1 and 2.2. These results clarify the strong dependency of the heat generation on the geometry of the particles as it depends on the eigenvalues of the Neumann-Poincar´e operator. In section 5 we present numerical examples of the temperature at the boundary of single and multiple particles. Appendix A is devoted to the asymptotic analysis of layer potentials for the Helmholtz equation in dimension two. We also include an analysis for the invertibility of the single-layer potential for the Laplacian for the case of multiple particles.

2

Setting of the problem and the main results

In this paper, we use the Helmholtz equation for modeling the propagation of light. This can be thought of as a special case of Maxwell’s equations, when the incident wave ui is a transverse electric or transverse magnetic (TE or TM) polarized wave. This approximation, also called paraxial approximation [19], is a good model for a laser beam which are used, in particular, in full-field optical coherence tomography. We will therefore model the propagation of a laser beam in a host domain (tissue), hosting a nanoparticle. Let the nanoparticle occupy a bounded domain D b R2 of class C 1,α for some 0 < α < 1. Furthermore, let D = z + δB, where B is centered at the origin and |B| = O(1). We denote by εc (x) and µc (x), x ∈ D, the electric permittivity and magnetic permeability of the particle, respectively, both of which may depend on the frequency ω of the incident wave. Assume that εc (x) = ε0 ε0c , µc (x) = µ0 µ0c and that 0, 0. Here and throughout, ε0 and µ0 are the permittivity and permeability of vacuum. Similarly, we denote by εm (x) = ε0 ε0m and µm (x) = µ0 µ0m , x ∈ R2 \D the permittivity and permeability of the host medium, both of which do not depend on the frequency ω of the incident wave. Assume that εm and µm are real and strictly positive. The index of refraction of the medium (with the nanoparticle) is given by p p n(x) = ε0c µ0c χ(D)(x) + ε0m µ0m χ(R2 \D)(x), where χ denotes the indicator function. The scattering problem for a TE incident wave ui is modeled as follows:                    where

∂ ∂ν

c2 ∇u + ω 2 u = 0 in R2 \∂D, n2 u+ − u− = 0 on ∂D, 1 ∂u 1 ∂u − = 0 on ∂D, εm ∂ν + εc ∂ν −

∇·

(2.1)

us := u − ui satisfies the Sommerfeld radiation condition at infinity,

denotes the outward normal derivative and c =

2

√1 ε0 µ0

is the speed of light in vacuum.

We use the notation



∂ ∂ν ±

indicating ∂u (x) = lim+ ∇u(x ± tν(x)) · ν(x), ∂ν ± t→0

with ν being the outward unit normal vector to ∂D. The interaction of the electromagnetic waves with the medium produces a heat flow of energy which translates into a change of temperature governed by the heat equation [11]  ω ∂τ   − ∇ · γ∇τ = =(ε)|u|2 in (R2 \∂D) × (0, T ), ρC   ∂t 2π      τ+ − τ− = 0 on ∂D, (2.2) ∂τ ∂τ   − γ = 0 on ∂D, γ  c m   ∂ν + ∂ν −     τ (x, 0) = 0, where ρ = ρc χ(D) + ρm χ(R2 \D) is the mass density, C = Cc χ(D) + Cm χ(R2 \D) is the thermal capacity, γ = γc χ(D) + γm χ(R2 \D) is the thermal conductivity, T ∈ R is the final time of measurements and ε = εc χ(D) + εm χ(R2 \D). We further assume that ρc , ρm , Cc , Cm , γc , γm are real positive constants. Note that =(ε) = 0 in R2 \D and so, outside D, the heat equation is homogeneous. The coupling of equations (2.1) and (2.2) describes the physics of our problem. We remark that, in general, the index of refraction varies with temperature; hence, a solution to the above equations would imply a dependency on time for the electric field u, which contradicts the time-harmonic assumption leading to model (2.1). Nevertheless, the time-scale on the dynamics of the index of refraction is much larger than the time-scale on the dynamics of the interaction of the electromagnetic wave with the medium. Therefore, we will not integrate a time-varying component into the index of refraction. Let G(·, k) be the Green function for the Helmholtz operator ∆+k 2 satisfying the Sommerfeld radiation condition. In dimension two, G is given by i (1) G(x, k) = − H0 (k|x|), 4 (1)

where H0 is the Hankel function of first kind and order 0. We denote G(x, y, k) := G(x − y, k). Define the following single-layer potential and Neumann-Poincar´e integral operator Z k SD [ϕ](x) = G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D or x ∈ R2 , ∂D

and k ∗ (KD ) [ϕ](x) =

Z ∂D

∂G(x, y, k) ϕ(y)dσ(y), ∂ν(x)

x ∈ ∂D.

∗ respectively denote the single-layer Let I denote the identity operator and let SD and KD potential and the Neumann-Poincar´e operator associated to the Laplacian. Our main results in this paper are the following.

3

Theorem 2.1. For an incident wave ui ∈ C 2 (R2 ), the solution u to (2.1), inside a plasmonic particle occupying a domain D = z + δB, has the following asymptotic expansion as δ → 0 in L2 (D),    
δ3 i ∗ −1 i u = u (z) + δ(x − z) + SD λε I − KD [ν] · ∇u (z) + O ∗ )) , dist(λε , σ(KD 1

∗ ) denotes the spectrum of K∗ in H − 2 (∂D) and where ν is the outward normal to D, σ(KD D

λε :=

εc + εm . 2(εc − εm )

Theorem 2.2. Let u be the solution to (2.1). The solution τ to (2.2) on the boundary ∂D of a plasmonic particle occupying the domain D = z + δB has the following asymptotic expansion as δ → 0, uniformly in (x, t) ∈ ∂D × (0, T ),   δ 4 log δ bc ∗ −1 ∂FD (·, ·, bc ) τ (x, t) = FD (x, t, bc ) − VD (λγ I − KD ) [ ](x, t) + O ∗ ))2 , ∂ν dist(λε , σ(KD where ν is the outward normal to D and γc + γm , 2(γc − γm ) ρc Cc bc := , γc |x−y|2 Z tZ − ω e 4bc (t−t0 ) FD (x, t, bc ) := =(εc ) |u|2 (y)dydt0 , 0) 2πγc 4πb (t − t c 0 D |x−y|2 Z tZ − e 4bc (t−t0 ) bc [f ](x, t) := VD f (y, t0 )dydt0 . 0) 4πb (t − t c 0 ∂D λγ

:=

Remark 2.1. We remark that Theorem 2.1 and Theorem 2.2 are independent. A generalization of Theorem 2.2 to R3 is straightforward and the same type of small volume approximation can be found using the techniques presented in this paper. In fact, in R3 , the operators involved in the first term of the temperature small volume expansion are

FD (x, t, bc ) :=

bc VD [f ](x, t) :=

ω =(εc ) 2πγc Z tZ 0

∂D

Z tZ 0

e

D

e

|x−y|2 c (t−t0 )

− 4b

4πbc (t −

|x−y|2 − 4b (t−t0 ) c

4πbc (t −

2 0
3 |E| (y)dydt ,

t0 ) 0

2

0


3 f (y, t )dydt .

t0 )

2

Here E is the vectorial electric field as a result of Maxwell equations. A small volume expansion for E inside the nanoparticle for the plasmonic case can be found using the same techniques as those of [7]. Throughout this paper, we denote by L(E, F ) the set of bounded linear applications from 4

E to F and let L(E) := L(E, E) and let H s (∂D) to be the standard Sobolev space of order s on ∂D.

3

Preliminaries

3.1

Layer potentials for the Helmholtz equation in two dimensions

Let us recall some properties of the single-layer potential and the Neumann-Poincar´e integral operator [2]: 1

1

k : H − 2 (∂D) → H 2 (∂D), H 1 (R2 \∂D) is bounded; (i) SD loc 1

k [ϕ](x) = 0 for x ∈ R2 \∂D, ϕ ∈ H − 2 (∂D); (ii) (∆ + k 2 )SD 1

1

k )∗ : H − 2 (∂D) → H − 2 (∂D) is compact; (iii) (KD 1

k [ϕ], ϕ ∈ H − 2 (∂D), satisfies the Sommerfeld radiation condition at infinity; (iv) SD

(v)

k [ϕ] ∂SD k )∗ )[ϕ]. = (± 12 I + (KD ∂ν ± 1

We have that, for any ψ, φ ∈ H − 2 (∂D), ( i km [ψ], u + SD u := kc [φ], SD

x ∈ R2 \D,

(3.1)

x ∈ D,

2 √ √ with km = ω εm µm and kc = ω εc µc , satisfies ∇ · nc 2 ∇u + ω 2 u = 0 in R2 \∂D and u − ui satisfies the Sommerfeld radiation condition. 1 To satisfy the boundary transmission conditions, ψ, φ ∈ H − 2 (∂D) need to satisfy the following system of integral equations on ∂D  kc km  [φ] = −ui , [ψ] − SD SD  i (3.2)  1 1 I + (Kkm )∗
[ψ] + 1 1 I − (Kkc )∗
[φ] = − 1 ∂u .  D D εm 2 εc 2 εm ∂ν

The following result shows the existence of such a representation [4]. Theorem 3.1. The operator  2 1 1 1 T : H − 2 (∂D) → H 2 (∂D) × H − 2 (∂D)  

1 1 1 1 km ∗ kc ∗ km kc (ψ, φ) 7→ SD [ψ] − SD I + (KD ) [ψ] + I − (KD ) [φ] [φ], εm 2 εc 2 is invertible.

5

4

Heat generation

In this section we consider the coupling of equations (2.1) and (2.2), that is,  2   ∇ · c ∇u + ω 2 u = 0 in R2 \∂D,    n2     u+ − u− = 0 on ∂D,       ∂u 1 ∂u 1  − = 0 on ∂D,     εm ∂ν + εc ∂ν −       us := u − ui satisfies the Sommerfeld radiation condition at infinity,     ρc Cc ∂τ ω =(εc )|u|2 in D × (0, T ), − ∆τ =  γ ∂t 2πγ c c     ρ C ∂τ  m m   − ∆τ = 0 in (R2 \D) × (0, T ),   γ ∂t m      τ+ − τ− = 0 on ∂D,       ∂τ ∂τ   γm − γc = 0 on ∂D,   ∂ν + ∂ν −      τ (x, 0) = 0.

(4.1)

Under the assumption that the index of refraction n does not depend on the temperature, we can solve equation (2.1) separately from equation (2.2). Our goal is to establish a small volume expansion for the resulting temperature at the surface of the nanoparticule as a function of time. To do so, we first need to compute the electric field inside the nanoparticule as a result of a plasmonic resonance. We make use of layer potentials for the Helmholtz equation, described in subsection 3.1.

4.1

Small volume expansion of the inner field

We proceed in this section to prove Theorem 2.1. 4.1.1

Rescaling

Since we are working with nanoparticles, we want to rescale equation (3.2) to study the solution for a small volume approximation by using representation (3.1). Recall that D = z + δB. For any x ∈ ∂D, x e := x−z δ ∈ ∂B and for each function f defined on ∂D, we introduce a corresponding function defined on ∂B as follows η(f )(e x) = f (z + δe x). It follows that

k [ϕ](x) = δS δk [η(ϕ)](e SD x), B k )∗ [ϕ](x) = (Kδk )∗ [η(ϕ)](e (KD x), B

6

(4.2)

(4.3)

so system (3.2) becomes       

η(ui ) , δ
1 ∂ui δkc ∗ 1 η( ). I − (K ) [η(φ)] = − B 2 εm ∂ν

δkm δkc SB [η(ψ)] − SB [η(φ)] = − 1 1 εm 2 I

+

δkm ∗ (KB )




[η(ψ)] +

1 εc

(4.4)

Note that the system is defined on ∂B. δkm For δ small enough SB is invertible (see Appendix A). Therefore, δkm −1 δkc δkm −1 η(ψ) = (SB ) SB [η(φ)] − (SB ) [

η(ui ) ]. δ

Hence, we have the following equation for η(φ): AIB (δ)[η(φ)] = f I , where

AIB (δ) = fI

4.1.2

1 1 εm 2 I

= −


δkm −1 δkc δkm ∗ ) SB + ) (SB + (KB

1 ∂ui η( )+ εm ∂ν

1 1 εm 2 I

1 1 εc 2 I


δkc ∗ ) , − (KB


δkm −1 η(ui ) δkm ∗ ) [ ) (SB + (KB ]. δ

(4.5)

Proof of Theorem 2.1

To express the solution to (2.1) in D, asymptotically on the size of the nanoparticle δ, we make use of the representation (3.1). We derive an asymptotic expansion for η(φ) on δ to later compute δkc [η(φ)] and scale back to D. We divide the proof into three steps. δSB Step 1. We first compute an asymptotic for AIB (δ) and f I . Let H∗ (∂B) be defined by (A.3) with D replaced by B. In L(H∗ (∂B)), we have the following asymptotic expansion as δ → 0 (see Appendix A) δkm −1 δkc = PH0∗ + Uδkm (SeB + Υδkc ) + O(δ 2 log δ), ) SB (SB
1 1 δk ∗ ∗ I ± (KB ) = I ± KB + O(δ 2 log δ). 2 2 ∗ associated to the eigenvalue 1/2 (see Appendix A) and let Let ϕ0 be an eigenfunction of KB Uδkm be defined by (A.5) with k replaced with δkm . Then it follows that


1 ∗ I + KB Uδkm = Uδkm . 2 Therefore, in L(H∗ (∂B)),   1 1
1 1
∗ 1 I + I+ − KB PH0∗ + Uδkm (SeB + Υδkc ) + O(δ 2 log δ), AB (δ) = 2εm 2εc εm εc εm

7

and from the definition of Uδkm we get   1
1 1
∗ 1 SB [ϕ0 ] + τδkc 1 I + I+ − KB PH0∗ + AB (δ) = (·, ϕ0 )H∗ ϕ0 + O(δ 2 log δ). 2εm 2εc εm εc εm SB [ϕ0 ] + τδkm (4.6) In the same manner, in the space H∗ (∂B),  
∂ui 1 1 η(ui ) η(ui ) −1 I ∗ 2 e f = −η( ) + I + KB PH0∗ SB [ ] + Uδkm [ ] + O(δ log δ) . εm ∂ν 2 δ δ We can further develop f I . Indeed, for every x ˜ ∈ ∂B, a Taylor expansion yields ∂ui )(˜ x) = ν(˜ x) · ∇ui (δ x ˜ + z) = ν(˜ x) · ∇ui (z) + O(δ), ∂ν η(ui ) ui (δ x ˜ + z) ui (z) (˜ x) = = +x ˜ · ∇ui (z) + O(δ). δ δ δ

η(

The regularity of ui ensures that the previous formulas hold in H∗ (∂B). The fact that x ˜ · ∇ui (z) is harmonic in B and Lemma A.4 imply that 1 −1 ∗ [˜ x · ∇ui (z)] )PH0∗ SeB −ν · ∇ui (z) = ( I − KB 2 in H∗ (∂B). Thus, in H∗ (∂B),   ui (z) −1 i i e ∗ x · ∇u (z)] + Uδkm [ PH0 SB [˜ +x ˜∇u (z)] + O(δ) . δ

1 f = εm I

From the definition of Uδkm we get 1 f = εm I

−1 [˜ x PH0∗ SeB

! −1 [˜ x · ∇ui (z)], ϕ0 )H∗ ϕ0 (SeB ui (z)ϕ0 − + O(δ) . · ∇u (z)] + δ(SB [ϕ0 ] + τδkm ) SB [ϕ0 ] + τδkm (4.7) i

Step 2. We compute (AIB (δ))−1 f I . I −1 We begin by computing expansion of  an asymptotic  (AB (δ)) .

∗ The operator AI0 := 2ε1m + 2ε1c I + ε1m − ε1c KB maps H0∗ into H0∗ . Hence, the operator

defined by (which appears in the expansion of AIB (δ)) AIB,0 := AI0 PH0∗ +

1 SB [ϕ0 ] + τδkc (·, ϕ0 )H∗ ϕ0 , εm SB [ϕ0 ] + τδkm

is invertible of inverse (AIB,0 )−1 = (AI0 )−1 PH0∗ + εm

SB [ϕ0 ] + τδkm (·, ϕ0 )H∗ ϕ0 . SB [ϕ0 ] + τδkc

8

Therefore, we can write (AIB )−1 (δ) = I + (AIB,0 )−1 O(δ 2 log δ)


−1

(AIB,0 )−1 .

∗ is a compact self-adjoint operator in H∗ (∂B) it follows that [1, 5] Since KB

k(AI0 )−1 kL(H∗ (∂B)) ≤

c , dist(0, σ(AI0 ))

(4.8)

for a constant c. Therefore, for δ small enough, we obtain
−1 (AIB (δ))−1 f I = I + (AIB,0 )−1 O(δ 2 log δ) (AIB,0 )−1 f I =

=

(Se−1 [˜ x · ∇ui (z)], ϕ0 )H∗ ϕ0 ui (z)ϕ0 − B + δ(SB [ϕ0 ] + τδkc ) SB [ϕ0 ] + τδkc  ! δ −1 I −1 1 i x · ∇u (z)] + O PH0∗ SeB [˜ (A0 ) εm dist(0, σ(AI 0 ))


−1 I + (AIB,0 )−1 O(δ 2 log δ)

x · ∇ui (z)], ϕ0 )H∗ ϕ0 (Se−1 [˜ ui (z)ϕ0 1 −1 [˜ x · ∇ui (z)] + − B + (AI0 )−1 PH0∗ SeB δ(SB [ϕ0 ] + τδkc ) SB [ϕ0 ] + τδkc εm   δ O . dist(0, σ(AI 0 ))

∗ described in Lemma A.2 we can further develop the Using the representation formula of KB third term in the above expression to obtain −1 [˜ x (AI0 )−1 PH0∗ SeB

∞ −1 X [˜ x · ∇ui (z)], ϕj )H∗ ϕj (SeB

· ∇u (z)] = εm εm 1 2 + 2εc − εc − 1 λj j=1 i

=

∞ X j=1

−1 [˜ x · ∇ui (z)], ϕj )H∗ ϕj (SeB −1

[˜ x · ∇ui (z)], ϕj )H∗ ϕj − (SeB εm εm 1 + − − 1 λ j 2 2εc εc

!

−1 [˜ x · ∇ui (z)] +PH0∗ SeB ∞ X x · ∇ui (z)], ϕj )H∗ ϕj 1 (Se−1 [˜ −1 i e ∗ = PH0 SB [˜ x · ∇u (z)] + (λj − ) B . 2 λ − λj j=1

Using the same arguments as those in the proof of Lemma A.4, we have (ν · ∇ui (z), ϕj )H∗ 1 −1 (λj − )(SeB [˜ x · ∇ui (z)], ϕj )H∗ = , 2 λj − 12 and consequently, (AI0 )−1

1 −1 −1 ∗ −1 PH0∗ SeB [˜ x · ∇ui (z)] = PH0∗ SeB [˜ x · ∇ui (z)] + (λε I − KB ) [ν] · ∇ui (z). εm

9

Therefore, (AIB (δ))−1 f I =

(Se−1 [˜ x · ∇ui (z)], ϕ0 )H∗ ϕ0 ui (z)ϕ0 −1 − B + PH0∗ SeB [˜ x · ∇ui (z)] + δ(SB [ϕ0 ] + τδkc ) SB [ϕ0 ] + τδkc   δ ∗ −1 i . (λε I − KB ) [ν] · ∇u (z) + O dist(0, σ(AI 0 ))

δkc Step 3. Finally, we compute η(u) = δSB (AIB (δ))−1 f I . δkc From Appendix A, the following holds when SB is viewed as an operator from the space to H(∂B):

H∗ (∂B)

δkc SB = SeB + Υδkc + O(δ 2 log δ).

In particular, we have δkc [ϕ0 ] = SB [ϕ0 ] + τδkc + O(δ 2 log δ). SB

It can be verified that the same expansion holds when viewed as an operator from H∗ (∂B) into L2 (B). Note that the following identity holds   −1 −1 i (z)] ϕ e [˜ x · ∇u S Υ [˜ x · ∇ui (z)], ϕ0 )H∗ ϕ0 (SeB 0 δk c B −1 −1 − [˜ x · ∇ui (z)] = − [˜ x · ∇ui (z)]. + PH0∗ SeB + SeB SB [ϕ0 ] + τδkc SB [ϕ0 ] + τδkc Straightforward calculations and the fact that SB is harmonic in B yields  

δ2 δkc I −1 I i ∗ −1 i ˜ + SB λε I − KB [ν] · ∇u (z) + O δSB (AB (δ)) f = u (z) + δ x ∗ )) dist(λε , σ(KB in L2 (B). Using Lemma A.3 to scale back the estimate to D leads to the desired result.

4.2

Small volume expansion of the temperature

We proceed in this section to prove Theorem 2.2. To do so, we make use of the Laplace transform method [13, 16, 22]. Consider equation (4.1) and define the Laplace transform of a function g(t) by Z ∞ L(g)(s) = e−st g(t)dt. 0

Taking the Laplace transform of the equations on τ in (4.1) we formally obtain the following

10

system:                             

ρc Cc τˆ(·, s) − ∆ˆ τ (·, s) = L(gu )(·, s) in D, γc ρm Cm τˆ(·, s) − ∆ˆ τ (·, s) = 0 in R2 \D, s γm s

(4.9)

τˆ+ (·, s) − τˆ− (·, s) = 0 on ∂D, ∂ τˆ ∂ τˆ γm − γc = 0 on ∂D, ∂ν + ∂ν − τˆ(·, s) satisfies the Sommerfeld radiation condition at infinity,

ω =(εc )|u|2 , respectively, where τˆ(·, s) and L(gu )(·, s) are the Laplace transforms of τ and gu := 2πγ c and s ∈ C\(−∞, 0]. A rigorous justification for the derivation of system (4.9) and the validity of the inverse transform of the solution can be found in [16]. 1 Using layer potential techniques we have that, for any pˆ, qˆ ∈ H − 2 (∂D), τˆ defined by ( β p], x ∈ R2 \D, −SDγm [ˆ τˆ := (4.10) β γ −FˆD (·, y, βγc ) − SD c [ˆ q ], x ∈ D,

satisfies the differential equations q in (4.9) together with the Sommerfeld radiation condition. q ρ m Cm Here βγm := i s γm , βγc := i s ρcγCc c and FˆD (·, βγc ) :=

Z G(·, y, βγc )L(gu )(y)dy. D 1

To satisfy the boundary transmission conditions, pˆ and qˆ ∈ H − 2 (∂D) should satisfy the following system of integral equations on ∂D:  β β  q ] = −FˆD (·, βγc ), p] + SDγc [ˆ −SDγm [ˆ  (4.11)
∂ FˆD (·, βγc )  βγc ∗
1  −γm 1 I + (Kβγm )∗ [ˆ p] + γc − 2 I + (KD ) [ˆ q ] = −γc . D 2 ∂ν 4.2.1

Rescaling of the equations

Recall that D = z + δB, for any x ∈ ∂D, x e := is such that η(f )(e x) = f (z + δe x) and k [ϕ](x) SD k (KD )∗ [ϕ](x)

x−z δ

∈ ∂B, for each function f defined on ∂D, η

δk [η(ϕ)](e = δSB x), δk )∗ [η(ϕ)](e = (KB x).

We can also verify that FˆD (x, βγc ) = δ 2 FˆB (ˆ x, δβγc ), ∂ FˆD ∂ FˆB (x, βγc ) = δ (ˆ x.δβγc ). ∂ν ∂ν

11

Note that in the above identity, in the left-hand side we differentiate with respect to x while in the right-hand side we differentiate with respect to x ˜. To simplify the notation, we will use FˆB ˆ to refer to FB (·, δβγc ). We rescale system (4.11) to arrive at  δβ δβ  q )] = −δ FˆB , p)] + SB γc [η(ˆ −SB γm [η(ˆ 
∂ FˆB  δβγc ∗
1  −γm 1 I + (Kδβγm )∗ [η(ˆ ) p )] + γ − I + (K [η(ˆ q )] = −γ δ . c c B B 2 2 ∂ν δβ

For δ small enough, SB γc is invertible (see Appendix A). Therefore, it follows that h i δβ δβ δβ q )] + (SB γm )−1 δ FˆB . η(ˆ p) = (SB γm )−1 SB γc [η(ˆ Hence, we have the following equation for η(ˆ q ): AhB (δ)[η(ˆ q )] = f h , where
δβ
δβ δβ δβ + (KB γm )∗ (SB γm )−1 SB γc + γc − 21 I + (KB γc )∗ , h i
δβ ∂ FˆB δβ = −γc δ + γm 12 I + (KB γm )∗ (SB γm )−1 δ FˆB . ∂ν

AhB (δ) = −γm fh 4.2.2

1 2I

(4.12)

Proof of Theorem 2.2

To express the solution of (2.2) on ∂D × (0, T ), asymptotically on the size of the nanoparticle δ, we make use of the representation (4.10). We will compute an asymptotic expansion for η(ˆ q) δβγc on δ to later compute δSB [η(ˆ q )] on ∂B, scale back to D and take Laplace inverse. Using the asymptotic expansions of Appendix A the following asymptotic for AhB (δ) holds in L(H∗ (∂B)) AhB (δ) = Ah0 +O(δ 2 log δ), where Ah0

 =−



∗ 1 γc + γm I − γc − γm KB . 2

In the same manner, in H∗ (∂B), ∂ FˆB + γm ∂ν ∂ FˆB = −γc δ − γm ∂ν

 
1 δ 5 log δ ∗ e−1 ˆ I + KB SB [δ FB ] + O ∗ ))2 2 dist(λε , σ(KD  
1 δ 5 log δ −1 ˆ ∗ e−1 ˆ e I − KB SB [δ FB ] + γm SB [δ FB ] + O ∗ ))2 . 2 dist(λε , σ(KD   δ2 Here the remainder comes from the fact that FˆB = O dist(λε ,σ(K ∗ ))2 . f h = −γc δ

D

12

¯ We can further verify Note that ∆FˆB = η(L(gu )) − δ 2 βγ2c FˆB in B and ∆FˆB = 0 in R2 \D. that FˆB satisfies the assumption required in Lemma A.4. Thus we have
1 ∂ FˆB −1 ˆ ∗ e−1 ˆ SB [δ FB ] = −δ I − KB + Cu ϕ0 + γm SeB [δ FB ] + O 2 ∂ν   δ3 where Cu is a constant such that Cu = O dist(λε ,σ(K ∗ ))2 .



δ5 ∗ ))2 dist(λε , σ(KD

 ,

D

After replacing the above in the expression of f h we find that η(ˆ q ) = (AhB (δ))−1 f h = (λγ I −

ˆ

∗ )−1 [δ ∂ FB ] KB

∂ν

Cu γ m + ϕ0 + O (γc − γm )(λγ − 21 )



δ 5 log δ ∗ ))2 dist(λε , σ(KD

 ,

(4.13)

where λγ =

γc + γm . 2(γc − γm )

Finally, in H∗ (∂B),   ˆ Cu γ m δ 6 log δ δβγc δβγc ∗ −1 ∂δ FB ˆ η(ˆ τ ) = −δ FB −δSB (λγ I−KB ) [ ]− δS [ϕ0 ]+O ∗ ))2 . ∂ν dist(λε , σ(KD (γc − γm )(λγ − 12 ) B (4.14) It can be shown, from the regularity of the remainders, that the previous identity also holds in L2 (∂B). Using Holder’s inequality we can prove that 2

δβ

kSB γc [ϕ]kL∞ (∂B) ≤ CkϕkL2 (∂B) , for some constant C.  Hence, we findthat identity (4.14) also holds true uniformly on ∂B and 4 log δ δβγc f x) = O dist(λδ ε ,σ(K Cu δSB [ϕ0 ](˜ ∗ ))2 , uniformly in ∂B. Scaling back to D gives D

  ˆ δ 4 log δ βγc ∗ −1 ∂ FD (·, βγc ) ˆ τˆ(x, s) = −FD (x, βγc ) − SD (λγ I − KD ) [ ]+O ∗ ))2 . ∂ν dist(λε , σ(KD

(4.15)

Before we take the inverse Laplace transform to (4.15) we note that (see [22])
L K(x, ·, bc ) = −G(x, βγc ), where bc := ρcγCc c and K(x, ·, bc ) is the fundamental solution of the heat equation. In dimension two, K is given by |x|2

e− 4bc t K(x, t, γ) = . 4πbc t We denote K(x, y, t, t0 , bc ) := K(x − y, t − t0 , bc ). By the properties of the Laplace transform, we

13

have −FˆD (x, βγc ) = −

Z · Z

Z

0

K(x, y, ·, t , bc )gu (y)dydt

G(x, y, βγc )L(gu )(y)dy = L D

0

0

 .

D

We define FD as follows Z tZ FD (x, t, bc ) := 0

K(x, y, t, t0 , bc )gu (y)dydt0 .

(4.16)

D

Similarly, we have that for a function f Z · Z Z G(x, y, βγc )L(f )(y)dy = L − 0

∂D

0

0

0

K(x, y, ·, t , bc )f (y, t )dydt

 .

∂D

bc We define VD as follows bc [f ](x, t) := VD

Z tZ 0

K(x, y, t, t0 , bc )f (y, t0 )dydt0 .

(4.17)

∂D

Finally, using Fubini’s theorem and taking Laplace inverse we find that   δ 4 log δ bc ∗ −1 ∂FD (·, ·, bc ) τ (x, t) = FD (x, t, bc ) − VD (λγ I − KD ) [ ](x, t) + O ∗ ))2 , ∂ν dist(λε , σ(KD uniformly in (x, t) ∈ ∂D × (0, T ).

4.3

Temperature elevation at the plasmonic resonance

Suppose that the incident wave is ui (x) = eikm d·x , where d is a unit vector. For a nanoparticle occupying a domain D = z + δB, the inner field u solution to (2.1) is given by Theorem 2.1, which states that, in L2 (D), ∗ u ≈ eikm d·z 1 + ikm SD λε I − KD


−1


[ν] · d ,

and hence 2



|u| ≈ 1 + 2km < iSD λε I −


∗ −1 KD [ν]



2 ∗ −1 · d + km SD λε I − KD [ν] · d .

Using Lemma A.2, we can write ∗ SD λε I − KD


−1

[ν] · d =

∞ X (ν · d, ϕj )H∗ SD [ϕj ] j=1

λε − λj

and therefore, for a given plasmonic frequency ω, we have ∗ SD λε I − KD


−1

[ν] · d ≈

(ν · d, ϕj ∗ )H∗ SD [ϕj ∗ ] . λε (ω) − λj ∗

14

,

(4.18)

Here j ∗ is such that λj ∗ =