L. For the boundary condition of no force on the particle surface, e.g. a sphere in vaccuum, the solutions for a angular momentuml = 0and l6= 0are
where ais the sphere radius. Equation 4.4 and 4.5 can be solved numerically and therefore values for the oscillation periodTi can be derived. We can write
Ti = 2π·a
Si·νL (4.6)
with Ti(a)being the oscillation period, a being the particle diameter and Si = 2qia. Unfortunately Lamb’s theory can only be applied to spherical particles in vacuum or particles which are embedded in homogeneous media. This however doesn’t resemble the sample structure, which consists of a gold disc sitting on a glass substrate, that we use in our measurements. Therefore, in order to predict the elastomechanical behaviour, we have to employ FEM simulations. In the next section we will introduce our FEM model and compare it against Lamb’s theory.
4.2 Comsol modeling
The modelling of the mechanical properties of nano sized metal discs is done with Comsol Multi Physics[56], which is a FEM based program. To justify the usage of Comsol, the elastomechanical properties, specifically the evolution of the oscillation period T with radius a, of a gold sphere in vaccuum is simulated and compared to the solution given by [54]. To retrieve the dependence of the oscillation periodT1 the eigenmodes of a sphere octant are calculated. The sphere octant is chosen as a basis as it fascilitates the retrieval of symmetric eigenmodes. In order to narrow down the FEM solutions, the displacement of the sphere center is set to 0 and
dis-placement normal to the radial direction is prohibited. The eigenfrequency of the radial breathing mode for a given sphere radius a is then extracted from the Comsol solutions. A comparison of Comsol simulation and the Lamb solution for the first order breathing mode of equation 4.4 is shown in figure 4.2 .
r(nm)
Figure 4.2 Evolution of the oscillation period T1 of the first order breathing mode of a goldsphere with radius r in vacuum calculated from the navier equation(black) and com-sol(red). The scaling parameter for the first order breathing mode as derived from equation 4.4 isS1 = 2.954
The comsol results agree very well with Lamb’s theory. In addition Comsol al-lows to simulate different structure types which is why we use it as the standard modeling tool for the elastomechanical properties of metal particles throughout this work. To model the transient volume change in Comsol, a gold disc of 50nm height and variable radius is placed on a half sphere made out of silica glass. As a boundary condition, displacement normal to the gold-glass interface is prohib-ited. To model the lattice expansion that occurs 1ps after the excitation through a pump pulse, the initial displacement values for the disc at t = 0 were set to [uvw] = 10−3·[xyz]thereby mimicking an expanded disc structure. The half sphere of silica glass was assumed to have no prescribed displacement att= 0. In order to incorporate modal damping, an effective damping parameter, the rayleigh damp-ing parameter, has to be chosen. The Rayleigh dampdamp-ing parameter is a concept of solid mechanics that tries to approximate the frequency dispersion of the damping parameter through a parabola that is determined by the damping coefficients of two dominant modes, typically on the lower and upper boundary of the frequency spectrum [57]. The Rayleigh damping parameter is defined as
Σ = α
2ω + βω
2 (4.7)
were α and β are the damping coefficients of the two respective modes andω is the frequency. αandβ have to be chosen carefully because modes outside the
fre-4.2. COMSOL MODELING 49 quency interval of ωα and ωβ are strongly damped [57]. For the simulation the Rayleigh damping parameter was chosen such that the lifetime of the oscillation matches the experimental results. The experimentally determined lifetimes are on the order of 600ps. This yields α = 1011 1s and β = 10−10s. The mechanical prop-erties of gold were assumed with a young’s modulus of 70GPa, and a Poisson’s ratio of 0.40 which lies within the value range that can be found in literature [58].
Via the indirect MUMPS solver [59], the temporal evolution of the mechanically destabilized system is calculated in time increments of 200fs. Snapshots of the total displacement of the gold disc are shown in figure 4.3 .
t (ps)
10 50
0
displacement (10 nm)
50
-3
0 0.02
Figure 4.3 Modelling the elastomechanical response of a gold disc to an instantaneous temperature increase is done by assuming a linear diplacement[u, v, w]along the axis[xyz]. The temporal evolution shows the formation of the characteristic breathing mode.
In the next step the time dependent volume change of the gold disc is calculated via surface integration over the disc surface S of the displacement (u,v,w)
∆V = Z Z
(u, v, w)·δS (4.8)
An example data set for a gold disc of 50nm height and 70nm radius is shown in figure 4.4. After a steep drop an oscillation of ∆V sets in. The data set is cut to times bigger than tcutof f to isolate the oscillatory part of the trace. The fourier transformation yields a dominant fundamental mode at around 15GHz. This Com-sol model allows to calculate the frequency evolution of the fundamental breathing mode for various disc sizes. In a next step we compare the theoretical findings to measurements (see section 4.3).
-500
Figure 4.4 Timedependent volume change ∆V extracted from the Comsol data. After tcutof f a harmonic oscillation consisting of two modes sets in. In order to separate the eigenmodes from the steep drop at the beginning of the trace, the FFT is done for times >
tcutof f. The fundamental mode has a frequency of 15GHz and decays within several hundred picoseconds. The second mode has a frequency of 32GHz and is strongly damped.