[Paper Review] Particle-based simulations of steady-state mass transport at high Péclet numbers

Thomas Müller, Paolo Arosio+6 more|arXiv (Cornell University)|Oct 17, 2015

Groundwater flow and contamination studies3 citations

TL;DR

This paper presents a particle-based simulation method for steady-state mass transport at high Péclet numbers by initializing particles according to their flux distribution rather than random concentration. By propagating these flux-weighted particles through Monte Carlo trajectories and detecting their passage in predefined regions, the method accurately predicts downstream concentration profiles, showing excellent agreement with experimental microfluidic data using fluorescent colloids and validating the approach with a particle radius of 25 nm.

ABSTRACT

Conventional approaches for simulating steady-state distributions of particles under diffusive and advective transport at high Péclet numbers involve solving the diffusion and advection equations in at least two dimensions. Here, we present an alternative computational strategy by combining a particle-based rather than a field-based approach with the initialisation of particles in proportion to their flux. This method allows accurate prediction of the steady state and is applicable even at high Péclet numbers where traditional particle-based Monte-Carlo methods starting from randomly initialised particle distributions fail. We demonstrate that generating a flux of particles according to a predetermined density and velocity distribution at a single fixed time and initial location allows for accurate simulation of mass transport under flow. Specifically, upon initialisation in proportion to their flux, these particles are propagated individually and detected by summing up their Monte-Carlo trajectories in predefined detection regions. We demonstrate quantitative agreement of the predicted concentration profiles with the results of experiments performed with fluorescent particles in microfluidic channels under continuous flow. This approach is computationally advantageous and readily allows non-trivial initial distributions to be considered. In particular, this method is highly suitable for simulating advective and diffusive transport in microfluidic devices.

Motivation & Objective

• To address the computational inefficiency and failure of conventional particle-based Monte Carlo methods at high Péclet numbers due to poor initialization.
• To develop a simulation strategy that bypasses solving the Fokker-Planck equation by directly simulating particle trajectories from a flux-based initial condition.
• To enable accurate prediction of steady-state concentration profiles in complex geometries, particularly in microfluidic devices with Poiseuille flow.
• To validate the method experimentally using fluorescent colloids in microchannels and compare simulated profiles with measured data.
• To demonstrate the method’s robustness by using measured initial probability distributions as input, minimizing fitting parameters.

Proposed method

• Particles are initialized at t=0 and x=0 according to a flux-weighted distribution Φ(x=0,y,z) = C(x=0,y,z)·v(y,z)/v̄, where C is the initial concentration, v is the local fluid velocity, and v̄ is the average velocity.
• Each particle's trajectory is simulated using the Langevin equation, incorporating stochastic thermal motion and deterministic advection based on the flow field.
• The steady-state concentration profile is reconstructed by counting the number of particle time steps spent in predefined detection regions downstream.
• The method avoids the need for continuous particle injection by simulating all particles from a single initial time step, reducing computational cost.
• The approach accounts for the parabolic velocity profile in microchannels and eliminates detection bias near channel walls where particles move slowly.
• The simulation uses first-principles physics with only the particle radius as a free parameter, validated against experimental initial distributions.

Experimental results

Research questions

• RQ1 Can a particle-based simulation method accurately predict steady-state mass transport at high Péclet numbers where conventional Monte Carlo methods fail?
• RQ2 Does initializing particles according to their flux distribution rather than random concentration improve accuracy and efficiency in simulating advective-diffusive transport?
• RQ3 To what extent can experimentally measured initial probability distributions be used as input to predict downstream concentration profiles without fitting?
• RQ4 How well does the simulated particle distribution match experimental fluorescence microscopy data in microfluidic channels?
• RQ5 What is the quantitative agreement between simulated and measured diffusive broadening of a fluorescent colloid stream?

Key findings

• The method achieves excellent quantitative agreement between simulated and experimentally measured concentration profiles in microfluidic channels, with the best match obtained using the known particle radius of 25 nm.
• The normalized square error for the 25 nm particle simulation was 380 at 20 mm downstream, indicating agreement within approximately 20 times the experimental noise level.
• Simulations using particle radii of 10 nm or 40 nm yielded normalized errors of thousands to tens of thousands, confirming the sensitivity and accuracy of the method when the correct radius is used.
• The approach successfully captures the diffusive broadening of a fluorescent colloid beam over 80 mm of propagation, matching experimental images at multiple downstream positions.
• The method is computationally efficient and scalable, as the cost of calculating particle flux is weakly dependent on system geometry and boundary conditions.
• The validation demonstrates that the simulation is not a fit to data but a first-principles prediction using only the measured initial distribution and particle size.