BDHM (Brownian Dynamics with Hydrodynamics and Magnetic forces) combines magnetic dipole interactions and hydrodynamics on the point particle level with a heuristic repulsion model for the spatial expansion of the colloids to imitate the dynamics of the experiments described in

[1] Scalable high-throughput microfluidic separation of magnetic microparticles by Hongri Gu, Yonglin Chen, Anton Lüders et al.

By reproducing the qualitative behavior of the experiments in [1] with the heavily simplified toy model implemented in BDHM, we can show the generality of the underlying dynamic phenomena. All information and methods employed by BDHM are described in detail in the supplemental information of [1]. While BDHM can be used to repeat the corresponding simulations, we strongly encourage full reproduction using the information in said supplemental information.

• What BDHM is and what it is not
• Requirements
• How to build BDHM
• How to use BDHM
• What BDHM does
• How to interprete BDHM data
• Assumptions and simplifications
• Contents of the particular files
• Disclaimer
• Additional folders independent of BDHM

BDHM is a heavily simplified toy model that trades physical rigorosity for straightforward concepts to act as a proof of concept for the experimental results and the suggested system of [1]. It successfully reproduces the overall dynamic behavior of the experiments on a qualitative level and, thus, indicates that the found results are independent of the concrete realization of the system. However, BDHM does not generate precise quantitative results, and the corresponding numerical data should be interpreted qualitatively. Nevertheless, compared to the resolution of the experiments, BDHM can be used to get a rough idea of the collective behavior of the system on the particle level.

The program is written in C (C99 standard) and it can successfully be built with gcc 13.2 (which was tested through GitHub checks). The functionality was heavily tested for binaries made with gcc 6.3.0 and gcc 7.5.0. The data presented in [1] was mostly obtained through a binary obtained with icc (version 2020u4).

To compile BDHM, the library GSL must be linked. However, in the studies presented in [1], GSL is only used for random numbers to generate the random initial conditions. Replacing these initial conditions with fixed ones, the program can easily be adjusted so that GSL is not needed (when repeating the numerical calculations of [1]). The functionality of BDHM was tested for GSL 2.4 and GSL 2.7.1 and the simulations in [1] were performed with a binary where the latter was linked.

The source files of BDHM can be found in the folder BDHM in this repository. An example makefile is given in said folder. As usual, it can be used to compile the program by typing

make

in the console while being in the folder that contains the source files. If BDHM must be built "by hand", the particular C files can be compiled with

gcc -Wall -std=c99 -O3 -g -flto -c filename.c

first. Afterward, the main file BDHM3.out must be generated while linking everything together. This can be done by typing

gcc -Wall -std=c99 -O3 -g -flto -c lists.o inputdata.o distance.o init.o printdata.o hi.o bda.o force.o magnetics.o verletlist.o printdisplay.o flow.o partmanagement.o calc.o BDHM.o -o BDHM3.out -lgsl -lgslcblas -lm

in the console.

BDHM must be executed in the console using

./BDHM3.out

where the parameters of the simulation must be stated in a file called inputfile which is in the same folder as the binary. Note that this input file is "static" meaning that the parameters must always be stated at the same position. If inputfile is not present in the folder where BDHM is executed, the program exits with an error. To generate a new example input file, use the line

./BDHM3.out Setup

in the console. The input file has always the form of the following example:

Geometry	6.000000	3	10	4.000000 
InitBox	18.000000	7.000000	3.500000
N	100
dt	0.000000010000
StepNumber	100000	100
Magnetics	90.0 0.000000	1.000000	0.000000	9314.47
HiMode	1
VerletList	1.500000
Seed	100
Flow	1000000.0
ParticleManagement	20	10
StationaryMode 1
CalcVelocity	0	6
CalcParticlesPerPole	1	1000

The particular lines of such an input file and the meaning of the numbers are summarized in the following list:

• Geometry: This line defines the layout of the system. The first number is the lattice constant of the micromagnet grid (in units of the colloid diameter). The second and third numbers are the number of micromagnets in the x and y direction, respectively. The final number is the height of the simulation box (in units of the colloid diameter).
• InitBox: Here, the box is defined, where the particles are randomly placed at the start of the simulations. The three numbers are the measurements of the corresponding cuboid volume (in units of the colloid diameter).
• N: The corresponding value is the total number of colloids that can be in the simulation at the same time. For StationaryMode 1, N is the number of particles in the system. For StationaryMode 0, N is the maximum number of colloids that is possible.
• dt: Length of the simulation step (in units of the Brownian time given by the squared diameter divided by diffusion coefficient).
• StepNumber: This line of the input file defines the number of the performed simulation steps. The first value is the total number of simulation steps. The second number defines the distance of consecutive simulation steps that are saved in a generated output file (i.e., in the example, every 10th simulation step is saved).
• Magnetics: All parameters of the external magnetic field are given in this line. The first value is the magnitude of the magnetic field (in units of mT) and the following three values are the coordinates of the initial direction of the field. The last value is the frequency with which the field is rotating (in units 1 per Brownian time).
• HiMode: Using the numbers 0, 1, and 2, the algorithm for the integration of the particle dynamics can be adjusted. With the option 0, all hydrodynamic interactions between the colloids are neglected. Option 1 means that the magnetic forces result in hydrodynamic interactions but flow fields generated due to steric interactions are neglected. This is the default case used for all studies described in [1]. The last option 2 includes hydrodynamic interactions for all particle forces.
• VerletList: To reduce the computational cost, Verlet lists are implemented. The number in this line is the corresponding Verlet radius (in units of the colloid diameter).
• Seed: Random seed for the generation of the pseudo-random numbers.
• Flow: Here, the additional flow field can be tuned (when StationaryMode 0 is chosen). The corresponding value is the maximum velocity of the parabolic Poiseuille flow (in units of colloid diameter divided by the Brownian time).
• ParticleManagement: If StationaryMode 0 is chosen, then the particles are deleted and added to mimic the influx and outflux of colloidal particles in the experiments. This line controls the number of particles in the system. The first number is the initial number of particles with which the simulation starts (while N is the maximum number that can be present). The second number is the rate which dictates when new particles are added. In detail, it is the number of steps between two consecutive additions of new particles.
• StationaryMode: BDHM possesses two main modes. In the first mode, no additional flow field is added, which means the dispersion medium is "stationary" (i.e., StationaryMode 0). The second option (StationaryMode 1) adds a Poiseuille flow to the system.
• CalcVelocity: This option can be used to calculate the velocity of the "first" particles of a traveling "particle band". The first number enables the option (i.e., CalcVelocity 1). The second number chooses the position of the "finish line" (in units of micromagnets). After the first particles pass the "finish line", the simulation ends.
• CalcParticlesPerPole: If this option is chosen (i.e., CalcParticlesPerPole 1), the average number of particles per occupied micromagnet is computed. Additionally, an output file is written, where the absolute number of particles per pole is given. The second value corresponds to the number of steps after which the second file with the absolute pole occupation is written.

Note that the word and number separation in the input file should be done with the tab key (the program is only tested for this case).

All details of BDHM are given in the supplemental information of [1]. Using the parameters defined in the input file, BDHM generates an array of micromagnets that are separated by a repulsive wall from superparamagnetic colloids that are located in the "positive half volume" of the setup. Additionally, a rotating magnetic field is applied which influences the colloid dynamics and the "soft-magnetic" micromagnets.

BDHM possesses two main modes. The first mode (StationaryMode 1) corresponds to the studies without an additional flow field. Here, the function CalcVelocity can be used to determine the velocity of "particle bands" moving along the grid due to the rotation of the external field. The velocity calculation starts when the first 20 colloids pass the fourth line of micromagnets perpendicular to the movement direction. The "finish line" (i.e., the line of micromagnets perpendicular to the movement direction where the velocity calculation stops) can be defined in the input file. When the first 20 particles pass the finish line, the simulation stops and the velocity is written in the console. Also in this mode, the number of colloids per occupied micromagnet can be computed with the option CalcParticlesPerPole. The corresponding data are written in separate output files. During this mode, simplified periodic boundary conditions are applied in the x and y direction (see the supplemental information of [1]).

The second main mode (StationaryMode 0) corresponds to the studies in [1] with a Poiseuille flow field. Here, additional repulsive walls are added to the system (parallel to the first wall and perpendicular to the movement direction at the end of the array). These walls are not considered in the computations of the hydrodynamic interactions for simplicity. If this mode is chosen, new particles "spawn" at one side of the grid, and particles that leave the micromagnetic array are deleted (they are stored at a position outside of the array).

In both modes, the particle positions are written in an xyz file during a simulation. This file can be used in programs such as VMD (VMD: Visual molecular dynamics, W. Humphrey, A. Dalke K. Schulten, J. Mol. Graph., 1996, 14, 33–38) to visualize the dynamics.

Internally, the calculations are performed in units of the colloid diameter, the thermal energy kT, the Brownian time, and the diffusion coefficient of a colloid. Accordingly, the output data are also given in said units and must be converted to SI units for a comparison with the experiments. For instance, the particle band velocity is given in units of colloid diameter divided by the Brownian time and the particle positions in the xyz file are given in multiples of the particle diameter. Note also that the dimensions of the experimental setup can be used to approximate a connection between the maximum velocity of the Poiseuille flow and the volume flux measured in the experiments. A PDF file called Parameters.pdf can be found in the repository, which lists the assumptions and conversion factors used for the comparisons in [1].

To visualize the collective dynamics of the system, it is suggested to use external software that can interpret the xyz data generated by BDHM. For this, VMD: Visual molecular dynamics, W. Humphrey, A. Dalke K. Schulten, J. Mol. Graph., 1996, 14, 33–38 is utilized in the context of [1]. In addition to the colloid positions, BDHM stores the two "endpoints" of the elongated micromagents in the xyz file to indicate the magnetic array in the visualization with programs such as VMD.

In the following, we list some assumptions and simplifications made by BDHM. In future studies, a more rigorous model can be implemented by eliminating items from the list.

• Only hydrodynamic interactions on the point particle level are used. This means that the Oseen tensor and the Blake tensor are applied. In particular, the model for the hydrodynamic interactions (for instance, the Blake tensor) utilized in BDHM is fully taken out of the theory section of Sedimentation of colloidal particles near a wall: Stokesian dynamics simulations, R. B. Jones, et al., Phys. Chem. Chem. Phys., 1999, 1, 2131-2139. (see sections 2, 3 and the appendix of the given reference).
• The colloids and the micromagnets of the grid are modeled as point dipoles.
• The cuboidal micromagnets are modeled as spheroids and fitting demagnetization factors are applied to characterize the apparent susceptibility.
• The hydrodynamic interactions resulting from the steric particle repulsions are neglected in the studies of [1].
• When multiple walls are present (which is the case for the systems with the additional flow field), only the influence of the wall corresponding to the magnetic grid on the hydrodynamic interactions is taken into account.
• Gravity acting on the colloids is neglected.
• Brownian fluctuations are not considered.
• Contact friction between touching colloids or between the colloids and the wall, as well as hydrodynamic lubrication phenomena are not taken into account.
• The magnetic interactions are cut after a certain distance (see the supplemental information of [1]).
• Minimum image convention is used for the hydrodynamic and magnetic interactions when periodic boundary conditions are applied. Even for the corresponding long-ranged relations, only interactions with the nearest periodic image are considered for simplicity.
• Only the external magnetic field is influencing the magnitude and direction of the magnetic moments of the colloids and the microparticles.
• The magnitude of the rotating magnetic field is kept constant. This is in contrast to the experiments in [1], where the magnitude of the magnetic flux density is expected to vary if the field is generated by a rotating magnet.

Note that the toy model BDHM successfully reproduces the main features of qualitative collective behavior found in the experiments, even with all these simplifications. Nevertheless, if more physically rigorous results are needed, it is suggested first to consider replacing the point particle hydrodynamics with relations based on the Rotne-Prager-Yamakawa tensor (particularly, the Rotne-Prager-Blake tensor) and to include the hydrodynamic interaction that results from the steric interactions (which are approximated via the Weeks-Chandler-Andersen pair potential in BDHM). While interpreting the results of BDHM, it should be kept in mind that the hydrodynamic point particle model in combination with the heuristic repulsions (and the missing contact phenomena such as the contact friction) lead to unphysical behavior in the dynamics of the colloids as soon as some particles are close or in direct contact. In particular, the fast "vibration-like" fluctuations that result when multiple particles are assembled near a micromagent are attributed to such artifacts. The qualitative reproduction of the collective dynamics of the experiments in [1] is unaffected by this, showing the robustness of the observed phenomena.

BDHM consists of multiple C files. Here, a summary of the corresponding content of the particular files is given.

• BDHM.c: Main file which contains the simulation loop. It calls functions grouped in the other files.
• bda.c: Here, the integrators are implemented. How the particles are moved during each simulation step can be found in this file.
• calc.c: Functions for observables that can be computed during a simulation are given in calc.c.
• distance.c: Contains functions to compute the distance between particles.
• flow.c: Corresponds to the optional Poiseuille flow.
• force.c: All functions that correspond to the different forces between the components of the systems are summarized here.
• hi.c: The treatment of the hydrodynamic interactions of the system is implemented in this file.
• init.c: Initialization of the system at the start of the simulation is organized in init.c.
• inputdata.c: All functions in the context of the input file with the parameters are given here.
• lists.c: BDHM uses linked lists as a data structure. Here, the corresponding functions are implemented.
• magnetics.c: Contains the functions in the context of the magnetic properties of the systems such as the magnetic moments and the external field.
• partmanagement.c: This file contains functions to add or delete particles to the system.
• printdata.c: Here, the functions that write the output data are summarized.
• printdisplay.c: This file contains the text that is written to the console.
• verletlist.c: In verletlist.c, the functions to check if the Verlet lists must be updated are implemented. Verlet lists are a method to reduce the computation time of the simulations.

To understand how BDHM is working, it is suggested to start by reading the main simulation loop in the main C file BDHM.c. If details for the specific function used in the main loop are needed, it is then possible to navigate to the correct subfile that contains the source code of the specific commands. Also, if there are questions regarding individual parameters that are used through all functions in the various C files, it is always suggested to go back to the main file BDHM.c to check the global variables.

Note that BDHM is not an ongoing software project with active support. Instead, it is part of the supplemental information of [1] and reflects the state of the software used to obtain the corresponding numerical results. Hence, this program is not optimized for user accessibility but is grown with the needs of the underlying study. For any questions regarding [1] and the corresponding data, please contact the corresponding authors.

This repository also contains the folder Experiments_Solidworks_Files with CAD files corresponding to the experiments of [1]. Full credit and responsibility for these files belong to Dr. Hongri Gu and Yonglin Chen, who designed and performed the experiments of [1]. Information on the particular files is given in the README PDF, which can be found in the corresponding folder. For questions regarding the CAD data, please contact the corresponding authors of [1]. Note that the license declared in this repository applies to the simulation program BDHM only.