The present study reveals that the environmental refractive index, ne
, traced by surface-adsorbed and PE-coated fluorescent PS microbeads matches that of the fluid introduced into the microfluidic system very well down to a sensor diameter of about 9 μm. For a sensor size of 7.6 μm and below, some deviations from the expected behavior are observed, so that the present limit in size seems to be somewhere around 8 μm diameter. In the following, we will discuss the different potential causes for the deviation at smaller sizes in more detail to gain further insight into WGM sensing in particular in view of their importance for biosensing applications, which would benefit from a minimization of sensor dimension due to the 1/R dependence of the WGM shift upon molecular adsorption [12
The first interesting question is whether above determined bead refractive index of about ns
= 1.56 reflects really the physical condition of the outer bead volume or whether it functions simply as a free parameter that tunes the results into the desired range. This could happen in particular because of the presence of the PE coating, which has a lower refractive index than PS, nPE
= 1.47 [22
], and thus may contribute to an average index as experienced by the WGMs. The same arguments hold for the presence of the surface, which also has a lower index than PS (nglass
= 1.5255, as provided by the manufacturer, cf.
To rule out the influence of both PE coating and surface on the result obtained for the bead index, we acquired WGM spectra from freely floating and uncoated commercial yellow-green fluorescent PS beads with a nominal diameter of 10 μm. WGM spectra of these beads were obtained immediately after placing a small droplet of highly diluted particle suspension onto a microscope cover slip, and then focusing onto microbeads before they settled on the surface. The height of the beads above surface was several tens of micrometers as could be concluded from the z-axis movement needed for focusing. Thus obtained spectra were evaluated in the same way as those before and ns calculated by setting ne = 1.333, which gave a value of ns = 1.5638 ± 0.0047 in reasonable agreement with the results for the surface-adsorbed beads, which gave ns = 1.5584 ± 0.0042 (cf.
). It should be noted, however, that the surface-adsorbed beads studied are smaller in size on average and that there is a certain trend of decreasing index, ns, with decreasing radius observable (cf.
). In fact, the agreement between the average index obtained with the freely floating beads matches perfectly that obtained for the largest surface-adsorbed bead with R = 4.9 μm. From our experience, we know that smaller beads are more susceptible to the doping procedure, which might be reflected by the trend of decreasing bead index observed here. Therefore, the data seem reliable and in conclusion, an influence of both PE coating and surface on the bead index can be excluded.
The validity of thus determined bead indices, ns
, is further confirmed by the good match between the observed shifts in the mode positions and their theoretical curves based on Equation 1
, both shown in . The latter were calculated by varying only the environmental index, ne
, while fixing ns
and R to their respective values as obtained from the DI water spectra. The excellent agreement also for higher environmental indices, ne
, does not only corroborate the validity of the Airy approximations in the given size regime, but also the use of above determined bead indices, ns
. Therefore, once more we conclude that bead indices given in reflect the physical condition of the outer bead volume as it might be affected by the doping process and the presence of the fluorescent dye.
Another reason for the observed size dependence of the results is related to the effects of particle dimension on the evaluation procedure. First of all, because the free spectral range scales inversely with the bead radius, fewer and fewer modes fall into the spectral range of the detection system, thus reducing the overall information content available. To this lower number of modes add their broader widths (cf.
), which makes determination of their positions more uncertain and thus more susceptible to an erroneous interpretation. The effect of this worse primary data condition can be nicely seen from the dependence of the calculated bead radii, R, and mode numbers,
, which are shown in in dependence of the fluid index, nfl
. Typically, the spectrum evaluation on basis of Equation 2
is very stable with respect to the best-fit mode number,
. With increasing fluid index, however, the peak positions become less and less certain and the mode numbers start to fluctuate, typically by ±1. Since the mode number defines basically, how many wavelengths fit into a bead circumference, with every change in the mode number also the best-fit bead radius undergoes an alteration. This can be seen well in , where the arrows indicate a change in the mode number. For the largest bead under evaluation, this uncertainty sets in at a relatively high fluid index of about 1.38, for the second largest bead around 1.36, and for the smallest microbead radii studied around 1.34. Each time the mode number steps up or down, a change in radius can be observed, thus corroborating the expected correlation between
and R. For the smallest bead radius, R = 3.3 μm, the mode number changes frequently, indicating that here a limit in precision is reached due to the uncertainty in determining the mode positions. Over a wide range of fluid indices, nfl
, and bead radii, R, the determination of the environmental index of the microbeads, ne
, seems to be unaffected from this correlation between
and R (cf.
) and thus reveals that ne
and R are sufficiently decoupled to allow their simultaneous determination on basis of Equations 1
Figure 6. Bead radii of the different beads studied in dependence of the fluid index, nfl, of the DI water/glycerol mixture injected into the microfluidic flow cell as determined by the fitting procedure. The arrows indicate a step up or down of the corresponding (more ...)
Another cause for the deviation from linearity in the evolution of the environmental index, ne
, with the fluid index nfl
, could be related to the presence of the substrate and/or interfacial effects, such as an enrichment of glycerol in vicinity of the surface. An influence of the substrate on the bandwidths of WGM has already been discussed in the literature. Le Thomas et al
] calculated a maximum mode broadening of about Δλ = 3 nm for surface-adsorbed 6 μm PS beads in air (nsubstrate
= 1.5). In our case, the broadening should be even higher due to the lower index contrast, m = ns
, and the slightly higher substrate index of nglass
= 1.5255. We speculate that some of the above mentioned broad features we observe in the background of the larger beads studied (R ≈ 5 μm; cf.
), may have their origin in such coupling effects between surface and WGMs. Since we described these features by additional Voigt profiles during the fitting, the undistorted information about those WGM shifts exclusively caused by index changes could be extracted from the data. As we mentioned above, this was not possible for the smaller beads studied as well as the large beads at high fluid indices, where we used only a minimum number of Voigt profiles for description of the clearly discernible modes because of their significant broadening. Therefore, we cannot exclude that in these regimes the results of the fitting routine are somewhat influenced by WGM-surface coupling as described by Le Thomas et al
. and that this is one cause for the observed deviations from the expected behavior for bead sizes below 8 μm. This issue requires some further clarification in the future.
Also interfacial effects, such as an influence of the flow conditions or particular glycerol-surface interactions, are more likely to be experienced by smaller beads. They are, however, less likely because the SPR experiments performed on a flat surface bearing the same surface chemistry showed no deviation from linear behavior of the SPR response with the fluid index. It should also be noted that when changing the fluid in the microfluidic cell, great care was taken to remove the former DI water/glycerol mixture by rinsing the flow cell with a copious amount of DI water. Subsequently, the flow cell was flushed thoroughly with the next mixture before continuing the measurements. Therefore, poor fluid exchange cannot be the cause of the observed effects. Whatever the cause might be, interfacial effects should be independent of the way of data evaluation and therefore already be observable in the raw data. In fact, in , for the two smallest microbeads, there is some indication of a deviation of the mode positions from the expected dependency in particular for fluid indices close to that of water. This may be an indication that some effects take place during the initial exposure of the microbeads to the DI water/glycerol mixture. Why this can be observed only with smaller beads may be either addressed to their higher sensitivity to environmental changes or to said interfacial effects. Further work will be required to elucidate these questions in more detail.
This brings us to a more general error discussion. We found that the fit results based on Equation 2
for simultaneous determination of the parameters ν, m, and R are quite stable. The error bars shown in and correspond to the ±10% boundaries of the minimum deviation Δmin
found for the respective best fit, i.e.,
. These errors turned out to be very small, thus indicating that the evaluation procedure yields quite robust results. It should be noted that the best-fit deviation Δ, which is the sum of all deviations of 4 to 6 mode positions, is typically <0.25 nm, which means that the individual experimental modes deviate from their calculated counterparts by less than 0.042 nm. Therefore, the main errors introduced into the procedure are of experimental origin. Here, those imposed by insufficient knowledge of the exact mode positions are most crucial. It should be stressed that the number of free parameters in the algorithm is small and that with only few assumptions, e.g., by setting ne
= 1.333 for the environmental index in the case of a DI water environment, absolute values for ns
, ν, and R can be calculated for the entire parameter range studied. This implies, however, that also the mode positions need to be precisely determined on an absolute scale. It was for this reason that the spectral range of the detection system had been calibrated to a number of well-known laser lines (cf.
experimental section), which reduces the uncertainty mainly to the reproducibility by which the mechanical stage of the monochromator can be set to a certain wavelength. The manufacturer’s test sheet of our instrument gives a turret repeatability of ±0.067 nm and a drive repeatability of ±0.002 nm, so that we can safely assume ±0.07 nm as upper limit for the error of the absolute wavelength scale. To these errors adds that of the determination of the peak centers via the fitting of Voigt profiles (cf.
and ), which depends severely on bead radius and fluid index. A safe upper limit for this uncertainty is ±0.5 nm, yielding a total error in the peak determination of ±0.6 nm.
To get a clue on how crucial this uncertainty in the mode positions affects the determination of the different parameters, we calculated the corresponding partial derivatives,
, for ns
, and R, via Mie-Debye theory (the exact solutions was used here to exclude any influence of the Airy approximations on the error calculation) and from those the respective maximum errors
, where X is one of the parameters of interest and ΔλWGM
= 0.6 nm. The results, which are listed in for the smallest and the largest particle radii studied, indicate that also the experimental errors are quite small. This holds particularly for the bead index, ns
. Its large deviation from the known bulk value of PS of Δns
≈ 0.03 can obviously not been explained by a wrong absolute determination of the wavelength scale. Also, the deviations found in ne
particularly for small bead radii (cf.
) are beyond the boundaries of the experimental errors. It should be further noted that the errors under consideration are maximum errors. For certain restrictions in particle dimension and/or fluid indices, the errors can be much smaller. Up to fluid indices of 1.36, for example, the error in the mode positions is smaller than 0.025 nm irrespective of the bead radius and smaller than 0.005 nm for all bead radii except the smallest (R = 3.3 μm). The corresponding errors in the parameters ns
, and R are also given in for comparison.
Experimental errors for bead index, ns, environmental index, ne, and bead radius, R, in dependence of the uncertainty in determining the mode positions, Δλ, for the smallest and largest bead studied.