# Femtosecond Laser Irradiation of Plasmonic Nanoparticles in Polymer Matrix: Implications for Photothermal and Photochemical Material Alteration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Near-Field Effects

**Figure 1.**The modeling setup. A spherical gold nanoparticle of radius r

_{p}is irradiated by femtosecond laser pulses at the central wavelength λ. The complex refractive index of the particle, its specific heat, density and heat diffusivity are n

_{p}, C

_{p}, ρ

_{p}and χ

_{p}, respectively. The corresponding properties of the matrix are denoted by n

_{m}, C

_{m}, ρ

_{m}and χ

_{m}. For a complete list of notations, see in Appendix A1.

_{p}is on the order of 10 nm and the pulse length τ

_{pulse}lies in a range of 10 fs to 1 ps. We approximate the incident laser radiation as a quasi-monochromatic wave at the central wavelength λ.

^{(2)}is two-photon absorption cross-section of X, $\hslash \text{\omega}=2\pi \hslash c/\text{\lambda}$ is the photon energy and η is the quantum yield. The solution of Equation (1) reads:

_{X}

_{0}is the initial concentration of X. The fraction v = (N

_{X}

_{0}− N

_{X})/N

_{X}

_{0}of reacted species X is further referred to as the conversion.

_{inc}= I

_{0}f(t), where I

_{0}is the peak incident intensity and function f(t) is of a unity amplitude. If the dispersion on the particle size-scale is neglected, then the actual function $I(\overrightarrow{r},t)$ can be calculated by means of Mie theory [13]:

_{pulse}is the number of laser pulses applied.

_{m}. The complex refractive index of the metal particle is n

_{p}= n

_{p}' + in

_{p}".

_{p}= 10 nm and λ = 800 nm is presented in Figure 2a,b. Here, two field maxima are formed near the sphere opposite each other, following the dipole approximation. The locations of the maxima are directed by the incident wave polarization and, thus, are the same for all irradiated nanoparticles within the matrix. This gives an opportunity to induce anisotropy into laser-irradiated domains of the material. The results for r

_{p}= 35 nm and λ = 400 nm are presented in Figure 2c,d. Here, the dipole approximation is not applicable, because the thickness of the skin-layer becomes smaller than the diameter of the particle. Thus, the higher-order spherical modes become important, resulting in the shift of the maxima towards the direction of the incident wave propagation. The photochemical reaction in this case results not only in laser-induced anisotropy, but also in the laser-induced loss of the center of symmetry. This is important for the applications in nonlinear optics, in particular for the second harmonic generation. It is notable that such a modification can be performed using reasonable irradiation regimes and materials without an outstanding two-photon sensitivity (see Figure 2 caption). This allows for possible applications of commercially available materials and lasers for a single-step modification of large volumes of about 1 cm

^{3}.

_{e–ph}= 1–3 ps [2]. This is the period during which the particle is effectively heated, even if the actual laser pulse length is shorter. The temperature diffusion time over the particle is τ

_{diffus}= r

_{p}

^{2}χ

_{p}

^{−1}, where χ

_{p}is the heat diffusion coefficient, which is on the order of 1 cm

^{2}/s for many metals, including gold and silver. For r

_{p}= 10 nm, the value τ

_{diffus}= 1 ps, which is on the order of τ

_{e–ph}.

**Figure 2.**The conversion of species X near the gold nanoparticle (indicated by a white circle) irradiated by N

_{pulse}= 10

^{5}laser pulses of length τ

_{pulse}= 100 fs. The two-photon absorption cross-section is σ

_{X}

^{(2)}= 10 GM (1 GM = 10

^{−50}cm

^{4}·s). The quantum yield is η = 0.8, and the incident wave is linearly polarized along the x and propagates toward the z direction. For (

**a**) and (

**b**): r

_{p}= 10 nm, λ = 800 nm; the complex refractive index of gold is n

_{p}= 0.15 + 4.91i [15], the refractive index of polymethyl methacrylate (PMMA) is n

_{m}= 1.484 [16]; the incident beam intensity is I

_{0}= 5 × 10

^{8}W/cm

^{2}. For (

**c**) and (

**d**): r

_{p}= 35 nm, λ = 400 nm, n

_{p}= 1.47 + 1.954i [15], n

_{m}= 1.503 [16], I

_{0}= 2 × 10

^{9}W/cm

^{2}.

_{diffus}and t >> τ

_{e–ph}.

_{adiab}if the fraction of heat that leaks from the particle to the matrix during this period can be neglected. The heat penetration depth from the particle into the matrix by the moment t can be evaluated as $\sqrt{{\text{\chi}}_{m}t}$. Provided ΔT

_{p}is the temperature rise of the particle due to the laser pulse, the amount of heat transferred to the matrix is evaluated as ${Q}_{m}\approx {C}_{m}{\text{\rho}}_{m}\Delta {T}_{p}\cdot 4\text{\pi}{{r}_{p}}^{2}\sqrt{{\text{\chi}}_{m}t}$. The amount of heat transferred to the particle is Q

_{p}≈ C

_{p}ρ

_{p}ΔT

_{p}·4πr

_{p}

^{3}/3. The ratio is:

_{p}and C

_{m}are the specific heat capacities; ρ

_{p}and ρ

_{m}are the densities. Subscripts “p” and “m” denote the particle and the matrix, respectively. It is seen in Equation (5) that Q

_{m}/Q

_{p}<< 1 if:

_{p}ρ

_{p}/C

_{m}ρ

_{m}~ 1 for gold particles in a polymer matrix; thus, τ

_{adiab}≈ r

_{p}

^{2}χ

_{m}

^{−1}. The thermal diffusivity of the polymer, χ

_{m}, is on the order of 10

^{−3}cm

^{2}/s. Thus, for a 10-nm particle, τ

_{adiab}~ 100 ps.

_{m}<< χ

_{p}, the heat transfer from the particle to the matrix may be neglected during the time period needed to equalize the temperature inside the particle. The different time scales allow the solution of the problems of the particle heating and the heat diffusion from the particle to the matrix separately. Namely, the laser irradiation causes the uniform temperature rise in the particle by:

_{p}is the volume of the particle and $w(\overrightarrow{r},t)$ is the heat power density, which is given by:

_{p}

^{2}/n

_{m}

^{2}.

_{m}is the number density of chromophores in the matrix and η

_{T}is the quantum yield. Taking I(r, t) outside the particle from Equation (3), one can evaluate the temperature rise in the matrix as:

_{p}in the particle and the maximal temperature rise max(ΔT

_{m}) in the matrix due to two-photon absorption exhibit a pronounced plasmonic resonance, as is shown in Figure 3.

**Figure 3.**Temperature rise in the particle (black solid lines, left axis) and the temperature rise in the matrix due to the two-photon absorption (red dashed lines, right axis) after irradiation by a single laser pulse of duration τ

_{pulse}= 100 fs. The particle radius is either 10 nm (

**a**) or 50 nm (

**b**). The incident laser fluence is F

_{0}= 2 × 10

^{−4}J/cm

^{2}, which corresponds to the maximum field intensity I

_{0}= 2 × 10

^{9}W/cm

^{2}. The incident polarization is circular. The volume-specific heat values for the particle and the matrix are C

_{p}ρ

_{p}= 2.5 J·cm

^{−3}K

^{−1}and C

_{m}ρ

_{m}= 1.7 J·cm

^{−3}K

^{−1}, respectively. The dependencies of n

_{p}(λ) and n

_{m}(λ) for gold and PMMA are taken from [15,16], respectively. For the two-photon absorption, σ

_{m}

^{(2)}= 100 GM, N

_{m}= 6 × 10

^{20}cm

^{−3}and η

_{T}= 1 are employed.

_{p}= 100 K). Then, we calculate the maximum temperature rise max(ΔT

_{m}) in the matrix for that intensity. Additionally, we calculate the maximum field intensity that is reached near the particle.

_{m}

^{(2)}= 100 GM, and the number density of chromophores N

_{m}= 6 × 10

^{21}cm

^{−3}nearly matches the typical number density of monomer chains in the polymer. The other has realistic parameters: σ

_{m}

^{(2)}= 10 GM and N

_{m}= 1 × 10

^{20}cm

^{−3}. It is seen that direct heating of the matrix due to the two-photon absorption becomes advantageous in the longer wavelength part of the visible and near-IR spectrum. However, the field intensity reached near the particle in order to provide a temperature rise comparable to the one in the particle is above 10

^{11}W/cm

^{2}even in the case where the two-photon absorption is overestimated. This intensity is even higher for realistic matrices, overcoming the optical breakdown level.

**Figure 4.**The maximum temperature rise (dashed lines, right axis) and maximum field intensity (black solid lines, left axis) outside the particle reached when the temperature of the particle is raised by 100 K due to irradiation by a circularly-polarized single laser pulse of duration τ

_{pulse}= 100 fs. The particle radius is either 10 nm (

**a**) or 50 nm (

**b**). The volume-specific heat values for the particle and the matrix are C

_{p}ρ

_{p}= 2.5 J·cm

^{−3}K

^{−1}and C

_{m}ρ

_{m}= 1.7 J·cm

^{−3}K

^{−1}, respectively. The dependencies of n

_{p}(λ) and n

_{m}(λ) for gold and PMMA are taken from [15,16], respectively. The quantum yield η

_{T}= 1.

## 3. The Particle Heating Effect

_{X}

_{0}.

_{r}is the “room” temperature (the initial temperature of the surrounding matter or a thermostat), ΔT

_{p}is the initial particle temperature increment after the pulse (above the room temperature), r

_{p}is the radius of the particle, χ

_{m}is the heat diffusivity of the matrix and r is the radial coordinate. Here, we introduce the dimensionless parameter β = (3C

_{m}ρ

_{m})/(C

_{p}ρ

_{p}) (for gold nanoparticles in PMMA matrix, we have β = 1.821) and use the dimensionless variables: p = (β

^{2}/r

_{p}

^{2})χ

_{m}t is the dimensionless time and R = r/r

_{p}is the dimensionless coordinate (normalized by the nanoparticle radius). We suppose that only a small amount of dopant molecules are converted due to a single pulse. This allows us to neglect the effect of reaction enthalpy on the heat propagation process.

_{A}is the activation temperature and A is the reaction constant. We calculate the part of reacted molecules after one laser pulse. We suppose that this part is much less than unity, so that N

_{X}under the integral can be considered as a constant.

_{pulse}pulses, the conversion ν reads:

_{r}+ ΔT

_{p}g(R, p), where g(R, p) is the normalized function, so that g(R = 1, p = 0) = 1.

**Figure 5.**Evolution of normalized temperature g as a function of dimensionless time p in different points of the polymer matrix surrounding a gold nanoparticle: R = 1 (black line), R = 1.1 (blue line), R = 1.5 (red line) and R = 2 (green line).

_{A}/(T

_{r}+ ΔT

_{p}). It allows one to use the saddle-point method. Here, we can approximately calculate the integral G (see Equation (17)):

^{3}pulses per second. Each pulse raises the nanoparticle temperature to ΔT

_{p}. It was mentioned above that for small particles, ΔT

_{p}can be calculated using Equation (11). The laser pulse at the plasmon resonance wavelength (λ = 530 nm) with the fluence F

_{0}= 10

^{−4}J/cm

^{2}increases the temperature up to about 100 K. This value of fluence is available from commercial lasers and makes it possible to irradiate simultaneously an area of about 1 cm

^{2}. Thus, for our estimates, we will use ΔT

_{p}~ 100 K. It should be noted that when energetic femtosecond laser pulses are employed for large-volume processing, the beam power may be significantly higher than the critical value for self-focusing. However, the considered intensities (not higher than 10

^{9}W/cm

^{2}) are not sufficient to provide filamentation within a sample of reasonable thickness. Indeed, for the filamentation development, the retardation integral $B=\frac{2\pi}{\lambda}{\displaystyle \underset{0}{\overset{l}{\int}}I{n}_{2}}dz$ should be larger than unity. Here, l is the distance at which filamentation starts to develop and n

_{2}is the nonlinear refractive index. With the typical value n

_{2}≈ 10

^{−14}cm

^{2}/W and the intensity I ≈ 10

^{9}W/cm

^{2}, B = 1 will be reached at l ≈ 1.5 cm. This is an estimate for the possible thickness of the sample. Moreover, our studies [10,19] show that the presence of the plasmonic nanoparticles within the polymer matrix provides a significant negative input in the nonlinear refractive index, which essentially hinders the filamentation.

**Figure 6.**The integral G(R = 1) as a function of thermostat temperature for T

_{A}= 8000 K (black color) and T

_{A}= 5000 K (red color). The points show numerical calculations using Equation (16), and the line shows the approximation by Equation (19). The radius of a gold nanoparticle is r

_{p}= 20 nm.

^{13}s

^{−1}. An activation temperature range T

_{A}less than 12,000 K is typical of precursor molecules (see, for example, [20,21,22,23]). The typical heat diffusivity of polymers is χ

_{m}~ 10

^{−3}cm

^{2}/s. The radius r

_{p}of a nanoparticle is about 20 nm. The number of pulses N

_{pulse}can be up to 10

^{6}for a few minutes. Thus, the conversion near the particle surface, according to Equations (18) and (19), is:

_{r}(see Equation (19) and Figure 6). The complete analysis of this phenomenon is beyond the scope of this paper.

_{A}= 10,000 K, the increase in temperature by 100 K is not enough for the reaction. ΔT

_{p}= 150 K will transform about 10% of the dopant molecules, and ΔT

_{p}= 200 K is much more efficient. On the other hand, a large increase in temperature is not desirable, since it can lead to damage of the polymer matrix.

**Figure 7.**(

**a**) The conversion in the polymer matrix on the surface of a gold nanoparticle (radius r

_{p}= 20 nm) as a function of activation temperature, for different temperature increment ΔT

_{p}. (

**b**) The conversion in the polymer matrix on the surface of a gold nanoparticle (radius r

_{p}= 20 nm) as a function of temperature increment ΔT

_{p}, for different activation temperature T

_{A}. The thermostat temperature T

_{r}= 300 K. The number of pulses N

_{pulse}= 10

^{6}. The curves are calculated using Equation (20).

_{max}(R) = exp(−a(R − 1)

^{b}).

_{1}(R)/G

_{1}(R = 1). It is seen that a higher activation energy leads to a smaller volume of reaction. It is seen that the thermally-activated reaction can be effective enough at the very surface of the heated particle. However, the reaction domain is highly localized just near the particle, thus preventing the conversion of a large net amount of molecules. In order to convert remote molecules that are initially located at more significant distances from the plasmonic particles than the particle radius, one should use nonlocal effects, which are beyond the scope of this paper.

**Figure 8.**The distribution of normalized temperature g vs. normalized coordinate R at different moments of time: p = 0.1 (black line), p = 0.2 (red line), p = 0.5 (green line). Blue dots demonstrate the spatial distribution of the maximum temperature. The blue line is its approximation by g

_{max}(R) = exp(−a(R − 1)

^{b}). Inset: parameters a and b vs. coefficient β and their linear approximation by a = 2.06 + 0.41β and b = 0.61 − 0.034β.

**Figure 9.**Normalized number of reacted molecules G

_{1}(R)/G

_{1}(R = 1) according to Equation (21) for ΔT

_{p}= 100 K, T

_{r}= 300 K, T

_{A}= 10,000 K (green line), T

_{A}= 8000 K (black line) and T

_{A}= 5000 K (red line).

## 4. Conclusions

^{9}W/cm

^{2}and materials with a not outstanding two-photon sensitivity. This allows for possible applications of commercially available materials and lasers for a single-step modification of large volumes of about 1 cm

^{3}.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A1. List of Notations

$\overrightarrow{r}$ | Radius vector |

r | Radial coordinate |

x, y, z | Cartesian coordinates |

t | Time |

r_{p} | Radius of the particle |

V_{p} | Volume of the particle |

λ | Wavelength |

$\hslash $ | Planck constant |

ω | Angular frequency |

τ_{pulse} | Pulse length |

τ_{e–ph} | Electron-phonon coupling time |

τ_{diffus} | Temperature diffusion time |

τ_{adiab} | Time period after the pulse, during which the heat transfer from the particle to the matrix can be neglected |

T | Temperature |

ΔT_{p} | Temperature rise of the particle after a single laser pulse |

$\Delta {T}_{m}(\overrightarrow{r})$ | Temperature rise in the matrix after a single laser pulse due to the multiphoton absorption |

T_{A} | Activation temperature |

T_{r} | Thermostat temperature |

C_{p}, C_{m} | Heat capacities of the particle and the matrix |

ρ_{p}, ρ_{m} | Densities of the particle and the matrix |

χ_{p}, χ_{m} | Heat diffusion coefficients of the particle and the matrix |

n_{p}, n_{p}', n_{p}" | Complex refractive index of the particle, its real part (refraction index) and imaginary part (absorption index) |

n_{m} | Refractive index of the matrix |

ε, ε', ε" | Ratio of dielectric permittivities of the particle and the matrix, its real and imaginary parts |

N_{X} | Number density of species X |

N_{X}_{0} | Initial number density of species X |

ν | Conversion |

σ_{X}^{(2)} | Two-photon absorption cross section of X |

η | Quantum yield of two-photon photochemical destruction of species X |

N_{m} | Number density of chromophores that are involved in heating by two-photon absorption |

σ_{m}^{(2)} | Two-photon absorption cross section of chromophores |

η_{T} | Quantum yield of two-photon heating |

$I(\overrightarrow{r},t)$ | laser field intensity |

I_{inc}(t), I_{0}, f(t) | Incident field intensity, its amplitude, and normalized temporal shape function |

N_{pulse} | Number of laser pulses |

F_{0} | Incident fluence |

${\overrightarrow{E}}_{Mie}/{E}_{0}$ | Electric field magnification according to the solution of Mie problem for plane monochromatic wave |

$w(\overrightarrow{r},t)$ | Heat power density |

${w}_{2ph}(\overrightarrow{r},t)$ | Heat power density due to the two-photon absorption in the matrix |

Q_{m}, Q_{p} | Amount of heat transferred to the matrix and to the particle due to the laser pulse |

β | Dimensionless parameter, β = (3C _{m}ρ_{m})/(C_{p}ρ_{p}) |

p | Dimensionless time |

R | Dimensionless coordinate |

g(R, p) | Normalized temperature rise in the matrix due to the laser pulse |

A | Reaction constant |

G(R), G_{1}(R) | Fraction of destroyed species in a pulse |

a, b | Approximation parameters for the maximum temperature |

B | Retardation integral |

l | Distance at which the filamentation development starts |

n_{2} | Nonlinear refractive index |

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**MDPI and ACS Style**

Smirnov, A.A.; Pikulin, A.; Sapogova, N.; Bityurin, N.
Femtosecond Laser Irradiation of Plasmonic Nanoparticles in Polymer Matrix: Implications for Photothermal and Photochemical Material Alteration. *Micromachines* **2014**, *5*, 1202-1218.
https://doi.org/10.3390/mi5041202

**AMA Style**

Smirnov AA, Pikulin A, Sapogova N, Bityurin N.
Femtosecond Laser Irradiation of Plasmonic Nanoparticles in Polymer Matrix: Implications for Photothermal and Photochemical Material Alteration. *Micromachines*. 2014; 5(4):1202-1218.
https://doi.org/10.3390/mi5041202

**Chicago/Turabian Style**

Smirnov, Anton A., Alexander Pikulin, Natalia Sapogova, and Nikita Bityurin.
2014. "Femtosecond Laser Irradiation of Plasmonic Nanoparticles in Polymer Matrix: Implications for Photothermal and Photochemical Material Alteration" *Micromachines* 5, no. 4: 1202-1218.
https://doi.org/10.3390/mi5041202