Solving the World’s Toughest CFD Problems

## Optimization of Magnetic Blood Cleansing Microdevices

a Department of Chemical and Biomolecular Engineering, University of Cantabria, Spain
b State University of New York at Buffalo, Buffalo (NY), USA

## Separation of toxins with magnetic particles. Why is it so important?

The use of magnetic particles has recently expanded for a process known as detoxification in which different toxins are extra-corporeally captured from the bloodstream of intoxicated patients. The detoxification of biofluids is the most conceivable treatment in a high number of clinical conditions, some of them associated with high mortality rates such as sepsis. This is a lethal disease caused by a microbial infection that spreads through the bloodstream to overwhelm the body’s defenses. It represents the primary cause of death in hospital intensive care units, afflicting 18 million people yearly and accounting for over 200,000 deaths annually in the United States alone. The mortality rate can increase as much as 9% for every hour before the administration of the correct therapy. Hence, rapid elimination of the toxins is of paramount importance even in state-of-the-art hospital intensive care units.

We found that the limitations of the current treatments require the development of novel strategies such as the use of magnetic beads as toxin sequestering agents. Due to the magnetic properties of the particles, once the capture of the pathogens is complete, their separation from the patient’s blood can be performed in a continuous process using an external magnetic field generated by a permanent magnet. From the multiple magnetic microseparators developed in the last decade, we proposed the use of two-phase continuous-flow systems. These systems may be the best alternative because any flow restrictions and any degradation of the biofluid (i.e., non-specific entrapment of cells in the capturing region) are avoided, thus maintaining the efficacy and capacity of the system over time [1]. However, the optimization of these processes has been less studied and rational design is often lacking because of the complexity associated with their mathematical description. Therefore, we modeled the separation of magnetic beads from flowing blood streams inside a multiphase system with FLOW-3D in order to optimize the design of extracorporeal detoxification processes. In the proposed separator design, shown in Fig. 1, the beads are continuously injected through the upper inlet, deflected by the application of a magnetic gradient and collected into a flowing buffer stream. In order to achieve the efficient separation avoiding the mixing of the fluid phases, magnetic and fluidic forces were carefully studied and optimized. A detailed study of particle-fluid interactions as the beads are deflected is also provided.

## Modeling approach with FLOW-3D

The model for predicting the magnetophoretic particle transport inside the bioseparator shown in Fig. 1 consists of a CFD-based Eulerian-Lagrangian approach. We used the Lagrangian framework to model the bead dynamics, whereas the fluid transport, which is predicted by solving the Navier-Stokes equations, is calculated with an Eulerian approach. According to the Lagrangian approach, particles were modelled as discrete units and the trajectory of each one was estimated by applying classical Newtonian dynamics. Although different force contributions act on the particles during separation, we considered only the dominant magnetic and fluidic forces for predicting the bead trajectories under magnetic gradients generated by permanent magnets. We obtained different particle trajectories and thus, separation efficacies, by varying the distance between the lower wall of the channel and the top of the magnet, while keeping the same inlet velocities of the fluids (0.035 m·s-1 for the buffer and 0.01 m·s-1 for blood solution). More details of the modeling effort we developed can be found in our published work [1, 2].

## Particle magnetophoresis results

By varying the position of the magnet, we demonstrated that variable magnetic field gradients are generated, and thus, different separation efficacies were obtained. Figure 2 shows where the trajectories of the particles under different distances “d” between the magnet and the channel: For distances between 0 and approximately 1 mm, all the particles are separated independently of their original position at the inlet. For larger distances, the separation is incomplete due to the low magnetic forces. Medium to high magnetic forces are necessary for achieving complete particle separation. However, we demonstrated that high magnetic forces are undesirable for detoxification purposes due to the extreme acceleration of the particles that leads to perturbations of the flow patterns and to disruption of the fluid interface, as seen in Fig. 3. Therefore, medium magnetic forces appear to be optimum for this kind of system, because complete bead separation can be achieved while maintaining  blood integrity.

## Conclusions

In this work, we introduced a novel FLOW-3D model for predicting and optimizing the process of magnetic bead separation from blood in a multiphase continuous-flow microdevice. This model takes into account the dominant forces acting on the particles and can be used to study critical details of the separation process, including the trajectories of individual particles, the time required for the separation, and the perturbation of the blood/buffer co-flows. A critical element of this work is that we studied the interaction between two fluids flowing simultaneously in the device while taking into account the effects of particle-fluid interactions in the flow field. These issues are very important since the solutions should flow independently along the length of the channel and be separated at their respective outlets, avoiding any possible blood loss or dissolution. The methodology followed here provides a rational design guide since it can be used to predict particle separation by taking into account key operational variables and parameters. It applies generally for parametric analyses and optimization not only for blood detoxification processes, but also for other studies that involve multiple confined liquid phases inside microfluidic devices. Our future work will focus on the experimental analysis of the process using whole blood and the integration of the magnetic separation stage with the removal of the toxins in order to design novel detoxification processes.

### Acknowledgements

Financial support from the Spanish Ministry of Economy and Competitiveness under the projects CTQ2015-66078-R (MINECO/FEDER) and CTQ2015-72364-EXP/AEI is gratefully acknowledged. Jenifer Gómez-Pastora also thanks the FPI postgraduate research grant (BES-2013-064415). Edward P. Furlani gratefully acknowledges financial support from the U.S. National Science Foundation, through Award CBET-1337860.

### References

[1] Gómez-Pastora et al., Separation and Purification Technology2017, 172, 16–31.

[2] Gómez-Pastora et al., Journal of Physical Chemistry C2017, 121, 7466−7477.

## Phase Change

Phase change physics are applicable to phase-diagram-on-a-chip, a relatively new concept in the field of microfluidics. Generally speaking, phase diagrams are graphs used in the field of physical chemistry to show various phases of a substance with changing conditions of pressure, temperature, etc. When a phase diagram is directly generated on a micro-chip (as opposed to the traditional way of drawing it on a page from experimental data) it is called a phase-diagram-on-a-chip. FLOW-3D‘s phase change model is a powerful tool for phase change microfluidics applications by accurately predicting the formation and interactions of vapor bubbles and the surrounding liquid – the bubbles can respond dynamically to liquid motion, and can grow or shrink depending on the temperature and pressure of their surroundings.

FLOW-3D animation showing the temperature of the heated plate along with the creation, evolution and subsequent collapse of the bubble from multiple angles.

## Computational Analysis of Drop Formation and Detachment

This article was contributed by Jelena Dinic and Vivek Sharma, Department of Chemical Engineering, University of Illinois at Chicago, Chicago, IL

## Introduction and Problem Statement

The rapid, repeated, precise creation and deposition of droplets, printing or patterning of small features (say l = 10-3-1 mm), and the formation of thin films with controlled, uniform thickness by spraying, are of great importance to a variety of old and new industrial applications (1-5). The liquid transfer and drop formation/deposition processes involve complex free-surface flows and formation of columnar necks that undergo spontaneous capillary-driven instability, thinning and pinch-off (1-5). Despite the progress made using experimental, theoretical and one-dimensional simulation studies for analyzing drop formation and liquid transfer for simple Newtonian and inelastic fluids, mechanistic understanding of printing and spraying remains a challenge. The primary motivation for the present computation effort is to examine the possibility of using the volume-of-fluid (VOF) approach embedded in the FLOW-3D to obtain mechanistic understanding of pinch-off dynamics of Newtonian fluids. We show that our computational analysis captures the complex interplay of capillary, inertial and viscous stresses that determines the self-similar capillary thinning and pinch-off dynamics. For the drop formation and detachment of Newtonian fluids, we show that the self-similar neck evolution obtained from the computational analysis can be described using the universal scaling laws expected from theory and 1D simulations (1-7) as well as experiments (1, 2, 8-12). Our success in simulating such prototypical flows is a necessary step towards using FLOW-3D for careful computational analysis of the nonlinear dynamics underlying finite-time singularity, satellite drop formation as well as printability in more complex geometries, that are significantly harder to describe or study using 1D models and experiments.

## Modeling Approach and Parameter Space

Simulations of drop formation and detachment from a nozzle were performed using uniform mesh size in FLOW-3D by using the Surface Tension and Gravity models. Drop formation and detachment of a finite volume of fluid involves the interplay of capillarity, inertia, viscosity and gravitational-induced drainage within a neck that connects a pendant growing drop to a nozzle. In the simulation, a finite volume of a Newtonian fluid is issued from a stainless steel nozzle (${{D}_{0}}=2{{R}_{0}}=1.7\,\text{mm}$). Detachment of a newly formed drop occurs once the gravitational force overcomes surface tension force ($mg>2\pi \sigma {{R}_{0}}$). Simulations are divided into two groups to elucidate the dramatic influence of fluid viscosity: low viscosity fluids (e.g., water and glycerol/water mixtures with glycerol content <40% by weight) and high viscosity fluids (e.g. glycerol and glycerol/water mixtures with shear viscosity > 100x Water Viscosity). Properties of the fluids of both groups are listed in Table 1 and 2, respectively.

Simulation of low viscosity fluid (Fluid 2 in Table 1) dripping from a nozzle. Color variable is the velocity magnitude (units: cm/s) and velocity vectors are shown.

Simulation of high viscosity fluid (Fluid 8 in Table 2) dripping from a nozzle. Color variable is the velocity magnitude (units: cm/s) and velocity vectors are shown.

## Discussion of the Simulation Results

Drop formation and detachment was simulated using FLOW-3D for fluids listed in Table 1 and 2, and the neck shape and neck radius evolution over time were analyzed. The shape of the neck and the neck thinning dynamics of low viscosity fluids (see Figure 1) exhibits the characteristic self-similar, inertio-capillary thinning behavior, anticipated by experiments, potential flow theory and 1D simulations (1, 2, 6, 7, 13):

(1) $\displaystyle \frac{{R(t)}}{{{{R}_{0}}}}\approx 0.8{{\left( {\frac{\sigma }{{\rho R_{0}^{3}}}} \right)}^{{\frac{1}{3}}}}{{\left( {{{t}_{c}}-t} \right)}^{{\frac{2}{3}}}}$

Here R(t) is the instantaneous radius of the neck, R0 is outer radius of the nozzle, $\displaystyle \sigma$ is the surface tension, $\displaystyle \rho$ is the density of the fluid and tc is the pinch-off time. Likewise, the radius evolution datasets for these higher viscosity Newtonian fluids appear to show a linear decrease in neck radius with time, and the thinning dynamics follow Papageorgiou’s visco-capillary scaling (8, 9) described by the following expression:

(2) $\displaystyle \frac{R}{{{{R}_{0}}}}=0.0709\frac{\sigma }{{{{\eta }_{s}}{{R}_{0}}}}({{t}_{p}}-t)$

We find that the measured values of capillary velocity (ratio of surface tension and viscosity) are comparable to the values obtained using commercially-available instrument called Capillary Break-up Extensional Rheometer (CaBER) by McKinley and Tripathi (8), and to the capillary velocity computed using their nominal surface tension and viscosity.

FLOW-3D allows for visualization of velocity vectors in the neck during thinning which gives insight into a nature of the flow. In addition, it gives the possibility of determining the moment during thinning after which the flow field within the thinning neck after initially experiencing a combination of shear and extension becomes purely extensional as shown in Figure 1. Additionally, thinning dynamics of low viscosity fluids show a qualitatively different behavior compared to high viscosity Newtonian fluids (see Figure 2). The neck profile for a low viscosity Newtonian fluid, becomes self-similar in agreement with theory (6, 13), in frames leading to the pinch-off.

## Conclusions, Outlook and Ongoing Work

Our preliminary results show that FLOW-3D based computational analysis can be used for simulating prototypical free-surface flows underlying drop formation and detachment. We find that the simulated radius evolution profiles match the scaling laws and pinch-off dynamics that are experimentally-observed and theoretically-predicted for inviscid fluids as well as high viscosity Newtonian fluids.

In contrast with often-used 1D or 2D models, FLOW-3D allows a robust evaluation of the magnitude of the underlying stresses and extensional flow field (both uniformity and magnitude) and the visualization of the flow filed within the thinning liquid filament (see Figure 1 and 2, for example). Stream-wise velocity gradients associated with extensional flow field arise within the columnar necks undergoing capillary-driven thinning. In rheologically-complex fluids, extra elastic stresses as well as non-Newtonian shear and extensional viscosity dramatically alter the nonlinear pinch-off dynamics (2, 10-12). We are currently implementing constitutive models with viscoelasticity and non-Newtonian rheology into FLOW-3D to develop robust computational protocols for assessing processability of complex fluids.

### References

1. J. Eggers, Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865-929 (1997).
2. G. H. McKinley, Visco-elasto-capillary thinning and break-up of complex fluids. Rheology Reviews, 1-48 (2005).
3. B. Derby, Inkjet Printing of Functional and Structural Materials: Fluid Property Requirements, Feature Stability, and Resolution. Annual Review of Materials Research 40, 395-414 (2010).
4. O. A. Basaran, H. Gao, P. P. Bhat, Nonstandard Inkjets. Annual Review of Fluid Mechanics 45, 85-113 (2013).
5. S. Kumar, Liquid Transfer in Printing Processes: Liquid Bridges with Moving Contact Lines. Annual Review of Fluid Mechanics 47, 67-94 (2014).
6. R. F. Day, E. J. Hinch, J. R. Lister, Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80, 704-707 (1998).
7. J. Eggers, M. A. Fontelos, Singularities: Formation, Structure, and Propagation. (Cambridge University Press, Cambridge, UK, 2015), vol. 53.
8. G. H. McKinley, A. Tripathi, How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer. J. Rheol. 44, 653-670 (2000).
9. D. T. Papageorgiou, On the breakup of viscous liquid threads. Phys. Fluids 7, 1529-1544 (1995).
10. J. Dinic, L. N. Jimenez, V. Sharma, Pinch-off dynamics and dripping-onto-substrate (DoS) rheometry of complex fluids. Lab on a Chip 17, 460-473 (2017).
11. J. Dinic, Y. Zhang, L. N. Jimenez, V. Sharma, Extensional Relaxation Times of Dilute, Aqueous Polymer Solutions. ACS Macro Letters 4, 804-808 (2015).
12. V. Sharma et al., The rheology of aqueous solutions of Ethyl Hydroxy-Ethyl Cellulose (EHEC) and its hydrophobically modified Analogue (hmEHEC): Extensional flow response in capillary break-up, jetting (ROJER) and in a cross-slot extensional rheometer. Soft Matter 11, 3251-3270 (2015).
13. J. R. Castrejón-Pita et al., Plethora of transitions during breakup of liquid filaments. Proc. Natl. Acad. Sci. U.S.A. 112, 4582-4587 (2015).

## Microfluidic Palette – A Gradient Generator

Continuing our work with microfluidics modeling, simulating and validating gradient generation devices is our latest undertaking at Flow Science. Diffusion based gradients are an integral part of many complex biological processes. One example is how wounds heal due to chemotaxis, where the cells migrate along a chemical gradient. Over the past few years, various approaches to establish and study diffusive gradients have surfaced, but they all suffer their share of problems.

Atencia, et al., have proposed an innovative microfluidic gradient generator – the microfluidic palette – that attempts to overcome the known problems with previous approaches.

## Previous Approaches and Associated Problems

There are three principal approaches to establishing diffusive gradients: laminar flows, membranes and hydrogels, and free diffusion. Each has its merits, but, as mentioned, they also have accompanying troubles.

The standard approach for studying and establishing gradients in microfluidic devices involves the use of laminar flows. This approach is very simple, but shear stresses are generated due to convective flows. Shear stress may alter the cellular response. For example, biased cellular migration and asymmetrical mass transport may occur.

A more-recent development is to avoid the convective flows by establishing diffusive gradients include using rigid membranes and hydrogels. However, membranes and gels reduce the diffusion speed affecting the transient development of a gradient.

Finally, an approach has been developed that brings two fluid plugs in contact to allow free diffusion between them. The approach is however limited to 1-D flows only. Also, once the gradient is established, convective flows have to be used to modify the diffusive gradients, which brings one back to the earlier problem of shear stress generation in laminar flows.

In this blog, I will discuss the principle behind the new approach to diffusive gradient generation proposed by Atencia, et al., and present some results of FLOW-3D simulations of such effects.

## The Microfluidic Palette

The principle behind the microfluidic palette is decoupling the convective flow from diffusion without using membranes or gels, leading to the following advantages:

1. Delivery of material (cells or soluble material) without shear stress
2. Generation of overlapping gradients with different spatial locations
3. Dynamic control over the gradients

The designs of the microfluidic palettes proposed by Atencia, et al., are shown above. In the 1-D case, a mass balance in Convection unit 1 shows that matching the flow rates at Inlet 1 and Outlet 1 prevents flow through the main microchannel while allowing transfer through diffusion. The Convection unit 1 acts as a perfect source. The 2-D case is simply an extension of the 1-D case, with more than two convection units.

## FLOW-3D Simulations

In the 1-D microfluidic palette animation below, a clean decoupling of the convective cells from the main central microchannel can be seen be through the plotted streamlines. The streamlines are all restrained to the convection units only and not even a single one leaks out into the microchannel, indicating excellent decoupling of convection and diffusion. The evolution of source concentration can be seen in the plot, which becomes visibly constant by the end of the animation.

The 2-D microfluidic palette demonstrates a spatio-temporal control on the generated gradients. The source and sink are rotated at an angular velocity. Also, after every t seconds, the active access port is deactivated and the next port is turned on. To see the live status of the diffusion inside the chamber, three line probes are placed in the simulation (marked in red, blue and black, respectively, in the bottom right window of the simulation below).

## Comparison with the Experimental Results

FLOW-3D results are in good agreement with the experimental results in terms of the evolution of concentrations inside of the chamber. The images below show a snapshot in time for both the experimental results and the simulation results. Notice that the experimental results are normalized. Also, experiments use fluorescence intensity to indicate the concentration of the source. In the simulation, FlowSight’s line probes are used to study the concentration between the 3 access ports.

FLOW-3D’s good validation against the experimental results paves a way of simulating other types of gradient generators in the future. There are more gradient generators like non-linear gradients generators, which I will discuss in future blogs.

### References

Atencia J, Morrow J, Locascio L.E., The microfluidic palette: A diffusive gradient generator with spatio-temporal control, The Royal Society of Chemistry 2009

## Mass Particles and Acoustophoretics

FLOW-3D v11.2 is all set to be released in a few weeks with many improvements to existing models and new developments. In anticipation of this release, we are turning the focus of our blog series to new developments coming in this release, like the particle model, wave absorbing components, mooring lines and interactive geometry.

One of the major  developments is the particle model, which has been significantly improved and expanded in FLOW-3D v11.2. In fact there are so many new capabilities in the particle model, that I will be discussing it in multiple posts, starting with this one where I will discuss mass particles.

In the new model, particles are grouped into the following classes based on their primary functionality:

• Marker particles are simple, massless markers, best used for tracking fluid flow
• Mass particles represent solid objects such as sand grains or inclusions
• Fluid particles are made up of fluid and inherit fluid properties, including solidification
• Gas particles represent bubbles that change their size in response to temperature and pressure loads from the surrounding fluid
• Void particles are similar to the gas particles but their specific function is to represent and track collapsed void regions, which is useful, for example, to predict potential porosity defects created during mold filling in casting
• Mass/momentum source particles represent user-defined mass/momentum sources in the mesh
• Probe particles act as diagnostic devices that record and report solution quantities at their locations; they can be made of particles of any other class
• User particles can be customized through user-defined functions in the source code

## Mass Particles

Once the mass particle option is enabled in FLOW-3D, users have the option of setting various mass particle species with varying diameters and densities. Additionally, the dynamics of the mass particles can be controlled by properties like diffusion coefficient, drag coefficient, turbulent Schmidt number and restitution coefficient. Mass particles can even be assigned thermal and electrical properties.

The user can set multiple sources for particle generation, and each source can have a mix of all or some mass particle species defined earlier. Also, users can choose a random or uniform generation of particles and even define the velocity at which particles are generated from the source. Altogether, there is a lot of flexibility in how the user can use this powerful particles model.

## Acoustophoretic Particle Separation

Acoustophoretic particle separation is one of many applications where mass particles can be directly used. Acoustophoretic particle separation represents a modern and efficient way of removing a large amount of objects from solutions in a microfluidic channel. The ability to separate suspended solid objects from a microfluidic solution is an important tool in a wide range of fields including: healthcare (e.g., malignant cell removal), research (e.g., nanoparticle separation), industrial (e.g., sequestration of suspended solids) and environmental (e.g., water purification). In principle, particle separation is achieved by acoustic forces. Principally, these forces are a combination of pressure forces generated by a standing wave field, fluidic drag and impulse forces resulting from collisions of the particles with the channel walls when the amplitude of the oscillation is sufficiently large. Due to this combination of forces, particles that are involved in an acoustophoretic process can be separated depending on their size and density.

To the best of our knowledge there are very few numerical studies on the subject that take into account all the aforementioned force contributions. So, in this article I will present an all-inclusive method of acoustophoretic modeling using FLOW-3D. By taking advantage of FLOW-3D’s unique modeling capabilities, we can readily introduce mass particles inside the domain in a random fashion using the updated particle model and then vibrate the entire domain with a given length amplitude at a specified frequency. The microchannel oscillations along with the rest of our numerical simulation results, can be easily visualized using FlowSightTM and its improved non-inertial reference frame rendering capabilities.

## Process Parameters

For the purpose of this analysis, a computational domain defining a microchannel with a square cross-section with 100μm edges and a total length of 1mm was used. A total of 1148 particles were initially introduced in the entire computational domain in a random fashion. We opted to oscillate the entire microchannel at a constant frequency of 10Khz and at multiple amplitudes. The length of the amplitudes ranged from 3.125μm to 50μm. As a general rule, larger oscillation amplitudes require smaller time-step sizes in order to account for the rapidly varying temporal variable changes. Nonetheless, the total analysis time was less than 2h on a 32 core standalone workstation.

## Results and Discussion

As seen in Fig. 1, the pressure field varies depending on the phase of the oscillation. More specifically, in Fig. 1a, we observe a pressure front located at the bottom of the channel which occurs during maximum upward acceleration while in Fig. 1b, the pressure front is located at the top of the channel which occurs at the time of maximum downward acceleration. Both Fig. 1 results refer to the case of maximum amplitude where the maximum pressure was found to equal or exceed 2400 Pa (approximately 0.24 Atm).

The particle separation animation shows the effectiveness of the acoustophoretic particle separation method and highlights the contributing forces. Note that the particles are affected mostly by pressure and drag forces at low amplitudes, but when the length amplitude of the oscillation becomes similar to the size of the microchannel, the particles are forced into a single separation plane due to impulse forces caused by collisions with the top and bottom walls of the microchannel. It is important to note that the numerical results obtained with this modeling method appear to indicate a separation level exceeding 90% for an overall process time of less than 4ms.

Based on our preliminary analysis, we can conclude that acoustophoretic processes can be a very efficient method of particle separation in terms of time and energy required. And with the improved particle model in v11.2, FLOW-3D represents a very powerful option for modeling such processes due its abundance of physical models and enhanced rendering capabilities.

Stay tuned for the next blog, in which I will discuss the new capabilities and possible applications of fluid particles.