Double Effective Medium Model for the Optical Properties

Various groups have tried, with mixed success, to explain the optical properties of thin films formed from organically cross-linked gold nanoparticles (mainly for LBL films).20,28

The optical properties of the films can be described with a two-step EMA.33,39 Two steps are needed because three phases are present in the film: the gold particles, the molecules, and voids, and because of the way they are distributed. The presence of the voids could be vaguely seen in previous SEM, scanning tunneling microscopy (STM), and AFM images21,30 (see Ref. [29] for supporting information) but they were not properly considered in previous attempts to use EMA,28 which is the main reason why these treatments failed to describe the experimental data properly.

The main assumption made in EMA is that the size of the grains forming the composite has to be smaller than the wavelength of the light, in order to prevent retardation and scattering effects from contributing. This is a good approximation for a large number of composite materials as is evidenced by the successful application of EMA to many samples over several decades. One thing, though, that has happened over the decades in the use of EMA schemes, is that the implications of the nanostructure were not always considered properly. Some analyses only "worked" because of the generic mathematical features of EMA and the use of multiple layers or too many free parameters, so fitting parameters must be interpreted cautiously.

The great appeal of basic EMA schemes is their mathematical simplicity, which makes them easy to use. One phenomenon which can complicate EMA treatments of a composite is a strongly localized fluctuation of the field strengths. Two useful effects can result—nonlinear dielectric response and, related to this, enhanced sensitivity in molecular and biomolecular sensing. Nonlinear effects are not present within the classical Maxwell Garnett (MG) and Bruggeman (BR) EMAs but there are extended EMAs which consider such effects.40,41

One point to be careful about is the role of the average nano- and microstructure of the composite. This is the main differentiation between the basic EMA schemes and is sometimes not properly considered. The danger comes from the fact that this structural contribution is embedded within the EMA equations through key parameter values such as apparent (not real) depolarization factors. Thus, a thorough understanding about their link to topology is needed, with all schemes based on the same integral equation.26 It is therefore imperative to identify the microstructure correctly in order to apply the most appropriate EMA and reach physically meaningful conclusions.

The two main EMA schemes used are MG42,43 and BR.44 While the MG scheme considers small, well-defined inclusions within a host matrix and hence treats the two materials asymmetrically, the BR treats both materials symmetrically, i.e., as topologically equivalent. The types of structures that are described by those schemes are shown schematically in Figure 5.9.

This distinction is also important to understand the two-step model that we find applied to the SA gold system.39 As the gold particles are well-separated inclusions in a dithiol matrix, this subsystem (gold-dithiol, AuL) should be treated within the MG scheme. The scheme most applicable to the total film (void inclusions plus AuL) is less obvious. In principle, both MG with voids as inclusions and a cellular BR structure are possible but a careful examination of the SEM images and comparison between MG and BR fits suggest a structure, which is best treated with the BR scheme. The formulas for the two steps, assuming spherical grains or particles, are as follows: MG:

Gold spheres in linker medium Voids in effective medium A

(MG style)

Voids in effective medium A (BR style)

Effective medium A

Final effective medium B

Final effective medium C

FIGURE 5.9 Schematic of possible double EMA models, showing the conceptual differences between an MG and BR effective medium. The model which seems to be the most appropriate in our case is the one building the final effective medium C.

where «L and «Au are the dielectric functions of the dithiol linker molecules and gold nanoparticles, respectively, «AuL , that for the gold-dithiol composite found by solving Equation 5.1 and «* the final, three-phase composite dielectric function. fAuL and fVoid are the volume fractions of the gold particles in the linker-gold system and the voids within the gold-dithiol-void composite, respectively. If nonspherical particles or grains are considered, the factor of 2 in the denominator changes according to the depolarization factor of the grains, which is based on their aspect ratio.

The mean free path of the conducting electrons is affected by the small size of the gold particles due to surface scattering and this alters the dielectric function «Au to be used. The standard technique is to change the broadening parameter, namely the electron relaxation time of the Drude part of the dielectric function, to be size-dependent. This is done by subtracting the ideal Drude part from the experimental bulk dielectric function and adding the size-dependent one. This is explained in more detail in Ref. [45] and specifically applied to the case presented here in Ref. [39].

The final model for the films is then a single layer described optically by the double EMA on the substrate. Any unknown parameters are typically wavelength independent, and hence easily overdeter-mined (not free), but some prior knowledge is preferred for unambiguous good solutions. They can be fitted with a standard thin film optical properties program applied to the experimental data. At this point we would like to stress the importance of keeping the number of adjustable parameters small. For EMAs it is sometimes tempting to add further parameters (e.g., depolarization factor or a rough surface layer) to improve the fit. This is something to be avoided unless justified by independent structural analysis. The inclusion of too many free parameters means a good fit is possible, but the physical meaning is lost.

In our case three parameters have to be determined: the volume fractions fAuL and f,oid as well as the thickness of the film. The thickness was determined from extrapolation of cross-section analyses of some films (see Section 5.2) and the volume fraction fAuL was determined from the peak position in (k), which is the resonance position. The spectral position of this peak is very sensitive to fAuL ( + 0.005) and as the voids are not affecting it, precise values were calculated, which are summarized in Table 5.2. The only parameter left to be determined by a fit of our model to the ellipsometry data is fvoid. As the ellipsometry provides two sets of data, we used a combined fit for all homogeneous

TABLE 5.2 Volume Fractions from the Fit of the Optical Model to the Ellipsometry Data for the Three Different Cross-Linkers

C2-Dithiol Volume Fraction (%) C8-Dithiol Volume Fraction (%) C15-Dithiol Volume Fraction (%)

samples (four films, filtrated from 3 to 7.5 mL of solution, see Table 5.1) and calculated a single /void for each linker with eight sets of data. This results in very well-determined values for /Void, which are presented in Table 5.2.

These results are one of the first examples where an MG model has been shown to work well at high volume fractions. The usual assumption is that it fails above / = 0.4, but this is based on calculations on ordered arrays46'47 where multipoles occur at high / Simple arguments,48'49 however, indicate that if the following two conditions are fulfilled—(i) positional randomness is maintained for the nanoparticles and (ii) particles do not locally cluster or touch—then all but dipole terms cancel. Touching particles introduce significant absorption tails at long wavelength. It is thus clear from these data that in SA networks of metal-linkers the linker keeps particles apart and the overall assembly is random in 3D.

Figure 5.10 shows a graphical representation of the linker chain length dependence on the volume fractions. The dependence of /AuL on linker length can be explained very simply: An increase in the chain length increases the average separation of the gold particles. This, in turn, decreases the amount of gold in a given volume, hence the volume fraction and this is exactly what can be observed in the graph.

The /void dependence shows something which can be correlated to the SEM images in Figure 5.2, a higher porosity of the films for the shorter linkers, or equivalently, a compacting of the films for the longer molecules. A possible reason for this compacting, as noted earlier (see Section 5.3), could be the larger structural flexibility of the longer chains and increased internal stress.

EMA treatments work only if the composite acts as a homogeneous medium, i.e., the spatial average over induced local fields must damp out all local field fluctuations. This requires a minimum length scale and hence layer thickness. Some indications of the minimum scale needed to achieve a smooth

Number of CH2 groups (-linker length)

FIGURE 5.10 Volume fraction dependence of /AuL and /oid on the chain length of the linker molecule.

Number of CH2 groups (-linker length)

FIGURE 5.10 Volume fraction dependence of /AuL and /oid on the chain length of the linker molecule.

average for a given sample can be gauged from the SNOM data in Figure 5.8. One has to consider though that these images are only a 2D projection, rather than a true volume representation. This may result in overestimating the averaging length scale. Figure 5.8 also shows, after careful consideration of the different resolutions, that the topology, as recorded via the AFM, averages out over much shorter distances than the local fields, so one cannot use it as a direct guide to estimate scales appropriate for an EMA treatment. The 2D projection (SNOM image) suggests that we need scales for the optical averaging of greater than 1 |mm for the C8 linker and greater than 2.5 mm for the C15 linker. The reason why EMA works for thinner films is that the characteristic length of a volume is considerably smaller than that for a plane, if both consist of the same number of isotropically distributed building blocks. A good averaging can thus be achieved on a smaller scale if the structure is 3D instead of 2D. Our rough estimate in using the 2D SNOM data is that for C8 a thickness of 200 to 400 nm is sufficient, whereas for C15 500 to 1000 nm may be necessary for a good average. If EMA should work, the dielectric behavior cannot depend on the layer thickness. Structural changes during films growth could be one reason for such a thickness dependence, as it would affect the interaction between the constituent materials. But even for fixed structures, like in the SA gold case presented here, one has to be careful. Other examples where one has to be cautious include very thin films (O 5 nm), films with surface roughness, and monolayer structures especially if percolating metallic structures are involved. One other contribution which can prevent the use of EMA, especially in the case of metallic structures, is the excitation of currents on the film surfaces, as those are not considered in classical EMA schemes. Dielectric properties can then appear to depend on film thickness, as these surface currents or surface plasmon polaritons can couple across thin films and to incoming radiation in the presence of nanostructures.38,50 Our results thus indicate that macroscopic surface plasmons are not present on either surface of these films even though they conduct to some extent by tunneling across the linker molecules.24,25 Electron density thus is too low and tunneling too slow in the linkers to support propagating plasmon polaritons.

The double EMA should only be applied to the samples where the substrate is completely covered by the gold-dithiol-void composite. Thinner films could perhaps be described with an EMA on an individual cluster basis but a good agreement is not necessarily expected in this case. This can be seen in Figure 5.11, which shows a comparison between an MG and BR fit for a thinner film for the C2 and C8 linkers, which is not completely covering the substrate. The differences for the C8 linker fits are not very large and the agreement with the experimental data is actually quite good. This might be an indication that the coverage of the substrate surface is more complete for this linker, compared to C2. For the C2 linker case, the MG and BR results differ considerably, especially in the spectral position of the peak in the extinction function (k).

Although the samples presented here have been prepared in a different way to the more widely used LBL technique, we expect those films to share the main structural features. This is supported by the similarity in the optical properties of films created by the different methods20,28,39 and similar structures

FIGURE 5.11 Comparison between experimental ellipsometric dielectric data (solid line) and MG (dotted line) and BR (dashed line) EMA for a film in the low-coverage regime (filtrated volume: 1 mL, see Table 5.1).

measured with STM, SEM, and AFM21'30 (see Ref. [[29] for supporting information). Our model for the optical properties should therefore also be applicable to the LBL films, although different void volume fractions might be expected due to the different methods of synthesis. We do not expect the values of fAuL to change very much, as they are defined by the average spacing of the particles and hence the specific molecules used. If the same dithiols are used in LBL-produced films we would actually expect the values to be the same as ours. The peak positions in Refs. [28,30] (both for C9 linkers) for example are comparable to our C8 results.

Another point which is interesting from an optical property point of view is that although for almost all films the inclusion of the void phase within the BR EMA scheme yields better fits and is consistent with the microstructure, as both phases (gold-dithiol and voids) percolate, the MG scheme starts to be comparable and even slightly better for the thickest films of the C15 linker. This can be explained by a look at Figure 5.7, which shows a film created from 7.5 mL of solution with C15 as linker molecule. The globules of gold-dithiol matter start to get so compact that the void phase starts to become isolated and its percolation is "destroyed." At this stage the MG scheme is more appropriate as compared to the BR, which is exactly what the optical measurements and the analysis with our model suggest.

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