The Quasi-Static Limit

As for the quantum versus classical electrodynamics, QED description is of course the correct and complete theory to describe all the molecular plasmonics phenomena. Nevertheless, it requires the definition and the manipulation of quantities that are often not as intuitive as their classical counterparts. Moreover, classical electrodynamics is able to explain most of the molecular plasmonics phenomena, and, with a few expedients, even intrinsically quantum-mechanical phenomena such as spontaneous emission. Therefore, in the following we shall stick to a classical electrodynamics description of the system, and we refer the reader to other works for a QED treatment [60]. 

As mentioned above, the molecule is typically treated as a field source (i.e. as a charge density and charge current), while the metal nanoparticle is considered as a region of space with different dielectric properties with respect to the medium that hosts the molecule and the nanoparticle. 

Strictly speaking, the plasmonics nanoparticle-molecule coupling should be always described by such a set of equation. While it can be demonstrated that not all the components of E, B, H, D fields are independent, it is still necessary to solve for at least one vectorial and one scalar field. However, Eqs. (5.1-5.4) greatly simplify when the intrinsic spatial variations of EM fields are smooth on the scale of the studied systems, i.e. when the latter are much smaller that the wavelength of the free propagating light at the relevant frequencies. In this situation we can in fact assume a wavevector k = co/c — 0. Under this limit, a few terms in the equations can be disregarded, and they simplify to  

3 The Point-Dipole Model of the Molecule, and the Classical Metal Nanoparticle 

Among the three models mentioned in Sec. 5.1 for molecular plasmonics, that using the point dipole model for the molecule is the most apt to introduce molecule-electrod3mamics coupling concepts [110, 111]. In this model, the molecule is punctiform, and only its dipolar properties are considered. In particular, the molecule is considered to have the same properties as in the vacuum (i.e. same transition moments, same polarizability). 

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Stress-Strain Cycles of Filled Rubbers in the Quasi-Static Limit... 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

In view of an illustration of the viscoelastic characteristics of the developed model, simulations of uniaxial stress-strain cycles in the small strain regime have been performed for various pre-strains, as depicted in Fig. 47b. Thereby, the material parameters obtained from the adaptation in Fig. 47a (Table 4, sample type C60) have been used. The dashed lines represent the polymer contributions, which include the pre-strain dependent hydrodynamic amplification of the polymer matrix. It becomes clear that in the small and medium strain regime a pronounced filler-induced hysteresis is predicted, due to the cyclic breakdown and re-aggregation of filler clusters. It can considered to be the main mechanism of energy dissipation of filler reinforced rubbers that appears even in the quasi-static limit. In addition, stress softening is present, also at small strains. It leads to the characteristic decline of the polymer contributions with rising pre-strain (dashed lines in... 

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

A combination of the sequences in Table 1 with MAS is also necessary to average out the CSA interaction. Such a combination is not straightforward, given the possible interference effects from the simultaneous presence of both RF and MAS. Such problems forced the above sequences to be applied in the quasi-static limit, where the MAS period Tj. is much longer than the cycle time of the sequence... 

It is interesting to note that Eq. (1.377) and Eq. (1.379) coincide exactly with the square of Eq. (1.199) and Eq. (1.201), respectively, which describe the field-enhancements of a dielectric sphere in the quasi-static limit. This is a manifestation of the so called optical reciprocity theorem and it can be shown that it holds for arbitrary geometry [46] (see also Sec. 5.3.3). 

Moreover, by decreasing the shell thickness we can note (see Fig. 3.17) a red-shift of the piasmon peak and this in agreement with extinction spectroscopies measurements on Au core-shells [57, 73], In the quasi-static limit, where the size of the nanoshell is much smaller than the wavelength of light, the piasmon resonance energies are determined by the aspect ratio [78, 93]. As the aspect ratio is increased, the piasmon resonance shifts to longer wavelengths [94,95]. 

Finally, the whole system (molecule + metal nanoparticle) can be treated atomistically via TD-DFT or other quantum chemical methods. The interaction between the metal nanoparticle and the molecule are treated on the same foot as the intra-molecule and intra-nanoparticle ones. This method is therefore able to include much more than just the electrodynamics coupling, as it can include mutual polarization, chemical bonding, charge transfers (also in excited states). On the down side, at present this approach is limited to very small metal particles (a few tens of atoms, a few nm in size). Moreover, electrodynamics coupling is limited to the quasi-static limit, as standard molecular Hamiltonian includes only non-retarded Coulombic potential. Nevertheless, this method represents a fully ab initio approach to molecular plasmonics. 

However, it is important to remark that for more realistic cases, this overlap between absorption and enhancement is lost, and for complex shaped particles for particles arrays and even for spherical particles outside the quasi-static limit, the maximum absorption frequency of the plasmon and the maximum available total field may appear at quite different frequencies. This is shown for example in Fig. 5.3 for a silver spherical particles. 

We can do a step further and, in the quasi-static limit, express Ynr in terms of the molecular transition dipole. To this aim, let us apply Eq. (5.26) to the volume W comprised between the two dashed surfaces in Fig. 5.4. We get ... 

Let us first consider the quasi-static limit V 0. For X = 0, equation (4.8) reduces... 

Even without using numerical methods, one can analyse some physically sound limiting cases of the exact solution for the case of strong collisions (7.64). First of all, (7.64) evidently reduces to the results of Robert and Galatry in the quasi-static case, i.e. when tc — oo. An opposite limiting case of fast fluctuations... 

The approaches are divided into those which do not take frequency into account (except for relaxation phenomena) and therefore are "static" theories and those which take frequency into account, generally by a scattering approach. The latter, "dynamic" theories, when reduced to the low frequency or quasi-static limit, usually compare favorably with the static theories. Some approaches take multiple scattering into account and cannot be solved in closed form. These require elaborate computer number crunching techniques (5). 

Similarly, the effects of bubbles in viscoelastic materials were studied by preparing rubber samples containing microvoids. Microvoids were used in order to avoid the effects of bubble resonance and to compare theories in their quasi-static limit. 

It should be noted that the quasi-static case is a limiting situation of the corresponding dynamic problem when the value of c is large. 

An important consideration in a CRAMPS experiment is the interference between the two averaging processes, i.e., does the physical rotation of the sample by MAS impair the performance of the multiple-pulse sequence, the latter having originally been designed for static samples. Indeed, a low vr, i.e., less than 3 kHz, is used in a conventional CRAMPS experiment, such that, to a first approximation, the sample can be considered to be static during each cycle of the multiple-pulse sequence. In this so-called quasi-static limit, the multiple-pulse sequence can be considered to take care of the... 

This model is supposed to capture the two extreme limits of granular flow, which are designating the rapid shear and quasi-static flow regimes. In the rapid shear flow regime the kinetic stress component dominates, whereas in the quasi-static flow regime the friction stress component dominates [127]. 

As far as measurements are concerned the perturbation method of Buravov and Shchegolev provides reliable results not only in the quasi-static region but practically at least one order of magnitude above the kb =>f limit. Moreover, for the spheroid of arbitrary complex permittivity we have found the universal perturbation relation valid for arbitrary wavelength inside the sample. This generalized approach not only covers various approximations used until now, but also strictly determines the limits of their applicability. ... 

These also result even if the motion is unsteady, providing that ajxV and the other dimensionless terms remain finite in the limit R = 0 Equations (7) and (8) are then referred to as the quasi-static or quasi-steady Stokes equations. In this case the time variable enters the equations of motion only in an implicit form. The precise relationship between the solutions of Eqs. (7) and (8) and the asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers is discussed in Section III. 

Theoretical description of dendritic structures is an active but difficult topic because the length between branched points is close to or even smaller than the Kuhn length, above which the segments can be thought of as if they are freely jointed with each other. Most modeling or simulation work has focused on the quasi-static and rheological properties of hyperbranched polymers and dendrimers in shear flow [240, 244, 255-263]. Limited amounts of data were published for dendrimers and hyperbranched polymers within elongational flow field. 

It should be stressed that due to the limitations of the quasi-static regime, the film thickness dependence of the band positions in the IR spectra is correctly described only within the framework of the polariton theory. Therefore, Eqs. (3.21) and (3.23), which relate to pure surface modes, give an inadequate thickness dependence of the band positions in the spectra. The general dispersion relation has the following simple form for ihe vacuum-film-metal interface and /7-polarization [13] ... 

The material behavior above-described refers to the quasi-static response. However, elastomers subjected to real world loading conditions possess fluid-like characteristics typical of a viscoelastic material. When loaded by means of a stepwise strain, they stress-relax, i.e., the reaction force resulting from the application of an initial peak falls to an asymptotic value, which is theoretically reached after an infinite time [69]. Moreover, if an external force is suddenly applied, creep is observed and the strain begins to change slowly towards a limiting value. 

The system composed by the molecule and the metal nanoparticle can be seen as a giant supermolecule, in which both the molecule and the metal can be treated at the same, ab initio, level. As such, molecular plasmonics phenomena are interpreted as the results of the excitations of this supermolecular system. One obvious limitation of this approach is the size of the systems that can be treated even with nowadays computers, noble metal particles larger than tens of atoms represent a challenge. Further limitations include those inherent the chosen ab initio method and the fact that this supermolecular approach intrinsically assumes the quasi-static approximation. In fact, the electrons of the molecule and the metal... 

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