14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Quantification of Brownian motion under presence of
flow using Particle Diffusometry

D. Lee1, S. T. Wereley1∗

1 Purdue University, School of Mechanical Engineering, West Lafayette, USA
∗ wereley@purdue.edu

Abstract
Particle diffusometry (PD), a quantification method for the Brownian motion, is performed by recording
temporally sequential images and using correlation analysis to obtain an ensemble diffusion coefficient
for all particles captured in the imaging region (Clayton et al., 2017). PD is proven to be successful in
the detection of the waterborne pathogen V. cholerae in environmental samples using different imaging
techniques, including an inverted fluorescence microscope as well as a handheld hardware device operated
with a smartphone (Clayton et al., 2019; Moehling et al., 2020). Although we intend to use PD to calculate
diffusion coefficients in quiescent fluid, oftentimes unintentional fluid flows occur, creating measurement
error when calculating the diffusion coefficient. In previous work, recordings under the presence of flow
were discarded to avoid incorrect measurements of the sample. The diffusion coefficient is calculated as:

D =
S2

c −S2
a

16M2∆t
(1)

Experimentally recorded image frames are obtained at an inter-frame time ∆t. These frames are then parsed
into interrogation windows. Interrogation windows from an image taken at ∆t is correlated with itself to
obtain autocorrelation peak width Sa. Two separate image frames recorded at t and t+∆t are correlated to
obtain cross-correlation peak width Sc. The Sc broadens compared to the Sa due to the presence of Brown-
ian motion. The obtained peak widths can be then averaged spatially or temporally over number of image
frames, to lower the random uncertainty of the measurement. The diffusion coefficient is then calculated
using Eq. 1 (Olsen and Adrian, 2000), where M is the total magnification of the recording setup. In this body
of work, we modify our approach in measuring the diffusion coefficient in the presence of fluid flow.

In the absence of flow, the correlation peaks described above produce axisymmetric Gaussian profiles. How-
ever, in the presence of flow, the correlation peak center shifts to the direction of the flow—exactly the
principle behind PIV. If the flow also has gradients in it, the correlation peak stretches and rotates in the
direction of shear, no longer axisymmetric. Since the Brownian motion along each axis is independent
of others, proper identification of streamwise and cross-streamwise directions allow a generalized rotated
two-dimensional elliptical Gaussian function to be fit to the cross-correlation (Chamarthy et al., 2009). The
diffusion coefficient can be calculated from the cross-streamwise correlation peak width.

In this work, simulated Brownian motion images were generated with a frame rate of 15 frames per second
with a diffusion coefficient of 1× 10−12m2/s with a variety of conditions: 0 to 80 degrees rotation with
10-degree increments, 1 to 10 pixel/∆t flow velocity with 10 pixel/∆t increments, and three different flow
types (uniform, Couette, and Poiseuille flows). Once the correlation peaks were calculated, all the peaks
from each of the interrogation windows were averaged temporally over 100 frames image sets to reduce
statistical random errors. A 3-point Gaussian peak fit was applied in both x- and y-directions to determine
the sub-pixel maximum location for further flow analysis. To calculate the cross-streamwise peak width,
a total of 11× 11 pixel2 areas were fitted (in contrast to the typical 5-point PIV peak fit), and the ellipse
equations at e−1 height is calculated using the fitted Gaussian profile. Resulting diffusion coefficient values
of varying angles and velocities were averaged with respect to the induced velocity values.

As shown in Fig. 1(a), a 3-point Gaussian sub-pixel fit on a single pass correlation poses no difficulty in
estimating the flow velocity from the generated simulation videos. A larger measurement variation occurred



in the low flow velocity region (i.e. 1-4 pixel/s), where distinguishing between the flow velocity and the
Brownian motion of particles presents a greater margin of error. For the case of measuring particle diffusion
coefficient, presented in Fig. 1(b), the diffusion coefficient calculated in the presence of uniform flow show
minimal deviation from their simulated values, hence the smaller error. However, with non-uniform flows,
we see greater deviations from theoretical values due to the gradient of the flow. Overall, an increase in error
is observed as the flow velocity is increased. Analysis reveals that with the developed algorithm, analyzed
diffusion coefficients show an error greater than 10% when the flow velocity exceeds 8 pixel/∆t.

Experimentally, flows were generated using a syringe pump through a rectangular channel. As the flow
profile forms an elongated Poiseuille flow along the wider dimension of the channel, the focal point is
adjusted to ensure the recorded flow profile is that of a uniform flow. Simulation parameter for the uniform
flow is extended to 22 pixel/∆t to better compare the trend between the simulation and experiment results.
Due to the difficulty of obtaining even increment of volumetric flow, x-directional error bar is incorporated
to account for the variation. The resulting Fig. 1(c) shows the experimental data following the trend of the
simulation results. The experiment shows the analyzed diffusion coefficients results in an error greater than
10% when the flow exceeds 18 pixel∆t with the developed algorithm. These 18 pixels account for 17% of
overall interrogation window. In other words, the velocity is equivalent to 6.6µm/∆t, a significant movement
if we take into account the particle size of 470nm that is used for the experimental verification. This current
body of work shows a promising initial work toward measuring diffusion coefficients in the presence of fluid
flow.

(a) Flow Velocity (b) Diffusion Coefficient (c) Experimental Verification

Figure 1: Comparison of simulation results with analyzed values (a,b), and with experiment values (c).

References
Chamarthy P, Garimella SV, and Wereley ST (2009) Non-intrusive temperature measurement using mi-

croscale visualization techniques. Experiments in Fluids 47:159–170

Clayton KN, Lee D, Wereley ST, and Kinzer-Ursem TL (2017) Measuring biotherapeutic viscosity and
degradation on-chip with particle diffusometry. Lab Chip 17:4148–4159

Clayton KN, Moehling TJ, Lee D, Wereley ST, Linnes JC, and Kinzer-Ursem TL (2019) Particle Diffu-
sometry: An Optical Detection Method for Vibrio cholerae Presence in Environmental Water Samples.
Scientific Reports 9:1739

Moehling TJ, Lee D, Henderson ME, McDonald MK, Tsang PH, Kaakeh S, Kim ES, Wereley ST, Kinzer-
Ursem TL, Clayton KN, and Linnes JC (2020) A smartphone-based particle diffusometry platform
for sub-attomolar detection of vibrio cholerae in environmental water. Biosensors & Bioelectronics
167:112497

Olsen MG and Adrian RJ (2000) Brownian motion and correlation in particle image velocimetry. Optics &
Laser Technology 32:621–627. optical methods in heat and fluid flow