Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM)

Abstract

A three-dimensional nanoparticle tracking technique using ratiometric total internal reflection fluorescence microscopy (R-TIRFM) is presented to experimentally examine the classic theory on the near-wall hindered Brownian diffusive motion. An evanescent wave field from the total internal reflection of a 488-nm bandwidth argon-ion laser is used to provide a thin illumination field on the order of a few hundred nanometers from the wall. Fluorescence-coated polystyrene spheres of 200±20 nm diameter (specific gravity=1.05) are used as tracers and a novel ratiometric analysis of their images allows the determination of fully three-dimensional particle locations and velocities. The experimental results show good agreement with the lateral hindrance theory, but show discrepancies from the normal hindrance theory. It is conjectured that the discrepancies can be attributed to the additional hindering effects, including electrostatic and electro-osmotic interactions between the negatively charged tracer particles and the glass surface.

Uncertainty analyses

Experimental uncertainties have been estimated based on a single-sample experiment, where only one measurement is made for each point (Kline and McClintock 1953). Four pertinent uncertainties are presented: (1) uncertainty for incident ray angle θ; (2) uncertainty for the lateral (x–y) Brownian displacement measurements Δx or Δy; (3) uncertainty for the penetration depth z_p; and (4) uncertainty for the z directional relative displacement Δh.

1.1 Uncertainty for incident ray angle

The evanescent wave rim radius at the back-focal plane of the lens, R (http://www.olympusmicro.com), is given as R=fnsinθ, where the focal length of the TIRF objective lens f=3 mm and the refractive index of the incident cover glass medium n=1.515. A measurement function for the incident ray angle is defined as sinθ=R/fn=g(R, f, n). The Kline-McClintock analysis (Fox et al. 2004) for uncertainty w _g gives:

$$\begin{array}{*{20}l} {{w_{g} } \hfill} & {{ = \pm {\left[ {{\left( {\frac{{\partial g}}{{\partial R}}w_{R} } \right)}^{2} + {\left( {\frac{{\partial g}}{{\partial f}}w_{f} } \right)}^{2} + {\left( {\frac{{\partial g}}{{\partial n}}w_{n} } \right)}^{2} } \right]}^{{\frac{1}{2}}} } \hfill} \\ {{} \hfill} & {{ = \pm {\left[ {{\left( {\frac{{w_{R} }}{R}g} \right)}^{2} + {\left( { - \frac{{w_{f} }}{f}g} \right)}^{2} + {\left( { - \frac{{w_{n} }}{n}g} \right)}^{2} } \right]}^{{\frac{1}{2}}} } \hfill} \\ {{} \hfill} & {{ = \pm {\left[ {{\left( {\frac{{w_{R} }}{{fn}}} \right)}^{2} + {\left( {\frac{{w_{f} \sin \theta }}{f}} \right)}^{2} + {\left( {\frac{{w_{n} \sin \theta }}{n}} \right)}^{2} } \right]}^{{\frac{1}{2}}} } \hfill} \\ \end{array} $$

(18)

where w _R , w _f , and w _n are the uncertainties associated with the individual parameters of R, f, and n. Per the resolution limit provided by Olympus Inc., w _R =±0.025 mm, and both the focal length uncertainty, w _f , and the refractive index uncertainty, w _n , are assumed to be negligibly small. For the incident angle of 62°, the resulting uncertainty is calculated as $w_{\theta } = \pm 0.315^\circ$ after a conversion based on w _θ =sin⁻¹(w _g ).

1.2 Uncertainty for the lateral (x–y) Brownian displacement measurements

A reasonable estimate of the measurement uncertainty due to random error is plus or minus half of the smallest scale division, equivalent to 1 pixel, of the CCD camera. By taking the average pixel displacements as 3 pixel, the lateral displacement uncertainty is estimated to be $w_{x} = w_{y} = \pm 0.5\;{\text{pixel}}\; \cong \; \pm 71.7\;{\text{nm}}.$

1.3 Uncertainty for the penetration depth z_p

Considering Eq. 2:

$$z_{{\text{p}}} = \frac{{\lambda _{0} }}{{4\pi }}{\left( {n^{2}_{i} \sin ^{2} \theta - n^{2}_{t} } \right)}^{{ - \frac{1}{2}}} $$

the uncertainty equation for z_p is given as:

$$w_{{z_{{\text{p}}} }} = \pm {\left[ {{\left( {\frac{{\partial z_{{\text{p}}} }}{{\partial \lambda _{0} }}w_{{\lambda _{0} }} } \right)}^{2} + {\left( {\frac{{\partial z_{{\text{p}}} }}{{\partial n_{i} }}w_{{n_{i} }} } \right)}^{2} + {\left( {\frac{{\partial z_{{\text{p}}} }}{{\partial n_{t} }}w_{{n_{t} }} } \right)}^{2} + {\left( {\frac{{\partial z_{{\text{p}}} }}{{\partial \theta }}w_{\theta } } \right)}^{2} } \right]}^{{\frac{1}{2}}} $$

(19)

where the optical blue filter for the laser beam has a bandwidth of $w_{{\lambda _{0} }} = \pm 2\;{\text{nm}},\;w_{\theta } = \pm 0.315^\circ$ (first section of Appendix), and variations of refractive indices are neglected, i.e., $w_{{n_{i} }} = w_{{n_{t} }} = 0.$ The measurement uncertainty of z_p shows a significant increase with the incident ray angle θ approaching the critical value of θ_c=61.38° (Fig. 10). For example, the penetration depth uncertainty is estimated as $w_{{z_{{\text{p}}} }} = \pm 4.75\;{\text{nm}}$ for θ=65°, but increases to $w_{{z_{{\text{p}}} }} = \pm 69.47\;{\text{nm}}$ for θ=62°.

Fig. 10

Calculated uncertainty for the penetration depth (z_p) for different incident angles (θ) with θ_c=61.38° for the glass–water interface (n _i =1.515 for glass, n _t =1.33 for water)

1.4 Uncertainty for the line-of-sight (z) Brownian displacement measurements

The uncertainty of Δh can be estimated by applying the uncertainty analysis to the ratiometric intensity relation given in Eq. 6 as:

$$RaInt = \frac{{I^{1}_{N} {\left( {h_{1} ,\;R,\;c} \right)}}}{{I^{2}_{N} {\left( {h_{2} ,\;R,\;c} \right)}}} = {\text{e}}^{{{\left( { - \frac{{\Delta h}}{{z_{{\text{p}}} }}} \right)}}} $$

or equivalently:

$$\Delta h = - z_{{\text{p}}} \ln {\left( {RaInt} \right)} = z_{{\text{p}}} {\left[ {\ln {\left( {I^{2}_{{\text{N}}} } \right)} - \ln {\left( {I^{1}_{{\text{N}}} } \right)}} \right]} $$

(6a)

Thus, the uncertainty equation is obtained as:

$$w_{{\Delta h}} = \pm {\left[ {{\left( {\frac{{\partial \Delta h}}{{\partial z_{{\text{p}}} }}w_{{z_{{\text{p}}} }} } \right)}^{2} + {\left( {\frac{{\partial \Delta h}}{{\partial I^{2}_{{\text{N}}} }}w_{{I^{2}_{{\text{N}}} }} } \right)}^{2} + {\left( {\frac{{\partial \Delta h}}{{\partial I^{1}_{{\text{N}}} }}w_{{I^{1}_{{\text{N}}} }} } \right)}^{2} } \right]}^{{\frac{1}{2}}} $$

(20)

The elementary uncertainties of $w_{{I^{1}_{{\text{N}}} }} \;{\text{and}}\;w_{{I^{2}_{{\text{N}}} }}$ may be estimated from the statistical nature of the measured data. A statistical analysis was conducted for the particle tracking data for θ _i =62° and the resulting statistical properties are summarized in terms of the pixel gray level, as shown in Table 2. Thus, the elementary uncertainties for the image intensities is assumed to be equal to the width of a 95% confidence interval, i.e., $w_{{I^{1}_{{\text{N}}} }} = w_{{I^{2}_{{\text{N}}} }} = \pm 10.6835$ (pixel equivalency), the reference particle image intensity $I^{2}_{{\text{N}}}$ is assumed to have the maximum intensity of 220, and the arbitrary particle image intensity $I^{1}_{{\text{N}}}$ is assumed to have the average intensity of 165.94. Substituting these values into Eq. 20, the overall uncertainty for the z location measurements is estimated as $w_{{\Delta h}} = \pm 29.38\;{\text{nm}}$ for θ=62°. This R-TIRFM uncertainty also decreases with increasing incident ray angle, depicting a similar trend of the penetration depth uncertainty.

Table 2 Statistical properties of the analysis conducted for the particle tracking data for θ _i =62°