Hybrid algorithm for extracting accurate tracer position distribution in evanescent wave nano-velocimetry

Abstract

Evanescent wave nano-velocimetry offers a unique three-dimensional measurement capability that allows for inferring tracer position distribution through the imaged particle intensities. Our previous study suggested that tracer polydispersity and failure to account for a near-wall tracer depletion layer would lead to compromised measurement accuracy. In this work, we report on a hybrid algorithm that converts the measured tracer intensities as a whole into their overall position distribution. The algorithm achieves a superior accuracy by using tracer size variation as a statistical analysis parameter.

Appendices

Appendix 1: Stochastic simulations

To quantify the accuracy of the hybrid analysis algorithm, the true tracer distribution p(Z) is obtained from stochastic simulations. Ensemble numerical data, which consist of non-dimensional radii A, positions Z, and intensities I, were generated and analyzed in the MATLAB environment. In each simulation, A of each tracer in the ensemble was first randomly sampled from a normal distribution with a mean of 1 and a standard deviation of S. Next, each tracer’s Z-position is randomly sampled from the Boltzmann distribution,

$$p\left( Z|A\right) = p_0 e^{-U\left( Z,A\right) / k_{\rm B} \Theta },$$

(13)

in a limited range of $\left[ Z_{\rm min}, Z_{\rm max}\right]$ by the inverse cumulative distribution function method (Gentle 1998). The lower bound $Z_{\rm min} = A$ was set by the physical constraint that no particle can penetrate the solid wall. The upper bound $Z_{\rm max}$, on the other hand, was set at a position where the effects of particle–wall interactions become insignificant (i.e., $U(Z) \ll k_{\rm B} \Theta$). Here, the total potential energy consists of electrostatic and van der Waals energies, or $U\left( Z,A\right) = U^{\rm el} + U^{\rm vdw}$ where

$$U^{\rm el}\left( Z,A\right) = B_{\rm ps} e^{-K \left( Z-A\right) }$$

(14)

is the electrostatic potential energy of the tracer. $K^{-1} \equiv \kappa ^{-1}/a_0$ and $\kappa ^{-1}$ is the Debye length of the electrolyte solution suspending the tracers (Prieve 1999), while $B_{\rm ps}$ is calculated based on the formula presented by Oberholzer et al. (1997). The van der Waals potential energy $U^{\rm vdw}$ is

$$U^{\rm vdw}\left( Z,A\right) = -\frac{A_{\rm ps}}{6}\left[ \frac{2AZ}{Z^2-A^2} + \ln \left( \frac{Z-A}{Z+A}\right) \right],$$

(15)

where $A_{\rm ps} \approx 10^{-20}$ J is the Hamaker constant (Oberholzer et al. 1997). With randomly sampled radii A and positions Z in place, corresponding particle intensities I were calculated using Eq. (1). Data for $5 \times 10^7$ particles were generated to represent a large polydisperse ensemble of near-wall tracers. Simulation parameter values were chosen based on the physical conditions of the evanescent wave nano-velocimetry experiments reported.

A simulation validation study was performed for 50-nm-radius polystyrene tracers near a glass wall with $\delta = 150\hbox { nm}, \kappa ^{-1} = 4\hbox { nm}$, and $S = 0$, which represents monodisperse tracers. Again, $\widetilde{p}(Z)$ represents a tracer ensemble’s position distribution inferred from its apparent intensity by assuming every tracer has a radius $A = 1$ (i.e., size uniformity), while p(Z) is used to represent the true position distribution of the same tracers. Since $\widetilde{p}(Z)$ explicitly assumes tracer size uniformity, the stochastic simulation is validated if $\widetilde{p}(Z) = p(Z)$ for monodisperse tracers. As shown in Fig. 7, the perfect agreement between $\widetilde{p}(Z)$ and p(Z) validates our simulation method and the differences shown in Sect. 5 can be truly attributed to tracer size variations. Furthermore, in all simulated cases with $S > 0$, we observe that the stochastic simulations are in excellent agreement with results derived from probability theory reported by Wang et al. (2011) (Fig. 8).

Fig. 7

Comparison of $p(\widetilde{Z})$ and p(Z) of monodisperse tracers $(S = 0)$ under evanescent wave illumination with intensities $I \ge 0.1$

Fig. 8

Simulated tracer position distributions under different size variations. Markers represent histograms obtained from the stochastic simulations and are in excellent agreement with the theoretical predictions (solid lines of corresponding colors)

Appendix 2: Langevin simulations

Consider a near-wall Brownian particle with a radius of a from a tracer ensemble whose mean radius is $a_0$. The displacement of this particle in z-direction between time $t_i$ and $t_{i+1}$ can be described by the Langevin equation (Huang et al. 2009),

$$z_{i+1} - z_i = \frac{\hbox {d}D_z}{\hbox {d}z}\delta t + \frac{D_z}{k_{\rm b}\Theta }F_z\delta t + N\left( 0,2D_z\delta t\right),$$

(16)

where $\delta t = t_{i+1} - t_i \ll \Delta t$ is the computational time step size, $D_z$ is the near-wall hindered diffusion coefficient in the z-direction, $F_z$ is the sum of all external forces acting on the particle in the z-direction, and $N\left( 0,2D_z\delta t\right)$ represents the hindered Brownian motion of the particle. Equation (16) can be non-dimensionalized into

$$\begin{aligned} Z_{i+1}& = {} Z_i + \left. \frac{\hbox {d}\beta _z}{\hbox {d}Z}\right| _{Z_i}\delta \tau + H\left( Z_i\right) \cdot \beta _z\left( Z_i\right) \delta \tau \nonumber \\&\quad + N\left( 0,2\beta _z\left( Z_i\right) \delta \tau \right), \end{aligned}$$

(17)

where $\tau \equiv D_0t/a_0^2$ and $D_0$ is the unhindered diffusion coefficient of an isolated tracer with a radius $a_0$. $\beta _z\left( Z_i\right)$ is given by Bevan and Prieve (2000),

$$\beta _z\left( Z_i\right) \equiv \frac{D_z\left( z\right) }{D_0} = \frac{6\left( \frac{Z}{A}-1\right) ^2 + 2\left( \frac{Z}{A}-1\right) }{6\left( \frac{Z}{A}-1\right) ^2 + 9\left( \frac{Z}{A}-1\right) + 2}.$$

(18)

A particle–wall interaction coefficient $H\left( Z_i\right)$ accounts for the electrostatic force $F_{\rm el}$ and the van der Waals force $F_{\rm vdw}$, and

$$H\left( Z_i\right) \equiv \frac{F_{z}\left( Z_i\right) a_0}{k_{\rm B}\Theta },$$

(19)

where $F_{z}\left( Z_i\right) = F_{\rm el}\left( Z_i\right) + F_{\rm vdw}\left( Z_i\right)$. The magnitudes of $F_{\rm el}(Z) = \partial U^{\rm el}/\partial Z$ and $F_{\rm vdw}(Z) = \partial U^{\rm vdw}/\partial Z$ are obtained by differentiating the potential energy equations (14) and (15), respectively.

At each time step $\tau _i$, the particle’s intensity $I\left( Z, A\right)$ was obtained from Eq. (1). After each time step, H(Z) and $\beta _z(Z)$ are updated and the particle’ position and intensity information is recorded. The simulation progressed for $\varDelta T/\delta \tau$ steps, where $\varDelta T = D_0\varDelta t/a_0^2$ is the non-dimensional exposure time. The computational time step size $\delta \tau$ was chosen to be $2 \times 10^{-4}$, which is greater than the particle momentum relaxation time $(\sim\mathsf {O}\left( 10^{-6}\right) )$ and short enough to ensure numerical convergence. Langevin simulations were carried out for each of the $2 \times 10^5$ tracers generated in the stochastic simulations.

The combined effects of the tracer size variation and the camera exposure time are considered through a time-averaged tracer intensity over $\varDelta T$,

$$\bar{I} \equiv \frac{1}{\varDelta T}\int _0^{\varDelta T} I\left( t\right) \hbox {d}t.$$

(20)

Using Eq. (1), the true time-averaged position of a tracer with radius A is

$$Z\left( \bar{I},A\right) = A - \varDelta \ln \left( \frac{\bar{I}}{A^3}\right) ,$$

(21)

and the true tracer position distribution is denoted as $p\left( Z\right)$. If one incorrectly assumes tracer size uniformity (i.e., $A = 1$), then $Z = 1 - \varDelta \ln \bar{I}$, and $\widetilde{p}\left( Z\right)$ denotes the corresponding tracer position distribution.