Selective Handling of Droplets in a Microfluidic
Device using Magnetic Rails
Bruno Testeabcd, Nicolas Jamondabcd, Davide ferraroabcd, Jean-Louis Viovyabcd and Laurent Malaquinabcd.
a) Institut Curie, Centre de Recherche, Paris, F-75248, France. E-mail: laurent.malaquin@curie.fr;
Fax: +33 (0) 1 40 51 06 36; Tel: +33(0) 1 56 2468 45.
b) CNRS, UMR168, Paris, F-75248, France.
c) UPMC, F-75248, France.
d) PSL, F-75248, France.
ONLINE RESOURCES
Microfluidics and Nanofluidics
1 Evaluation of the magnetic force
⃗
The energy 𝐸𝑚 (J) of a point-like magnetic dipole or magnetic moment 𝑚
⃗⃗ (A/m2) in a magnetic flux density 𝐵
(T) is given by:
⃗ ) = −(𝑚
⃗)
𝐸𝑚 = −𝜇𝑜 (𝑚
⃗⃗ . 𝐻
⃗⃗ . 𝐵
(1)
with 𝜇0 (H/m) the magnetic permeability of free space and H the magnetic field (A/m).
The corresponding magnetic force Fm (N) is given by:
⃗⃗⃗⃗
⃗ ) ≈ (𝑚
⃗
𝐹𝑚 = −∇ (𝐸𝑚 ) = ∇ (𝑚
⃗⃗ . 𝐵
⃗⃗ . ∇ )𝐵
(2)
In the linear region of the magnetization curve, for a superparamagnetic microparticle suspended in a weakly
diamagnetic medium such as water, the magnetic moment can be written in the following form:
⃗⃗ = 𝑉𝑝 ∆𝜒 𝐻
⃗ =
𝑚
⃗⃗ = 𝑉𝑝 𝑀
𝑉𝑝 ∆𝜒
𝜇0
⃗
𝐵
(3)
According (2) and (3), the magnetic force for a single superparamagnetic particle can be expressed by the
following integral over the volume of the particle:
⃗⃗⃗⃗
⃗⃗ . ∇)𝐻
⃗ . 𝑑𝑣 = 𝜇0 Δ𝜒 ∫ (𝐻
⃗ . ∇)𝐻
⃗ . 𝑑𝑣
𝐹𝑚 = 𝜇0 ∫ (𝑀
𝑉𝑝
(4)
𝑉𝑝
An exact calculation of the magnetic forces acting on a bead cluster is difficult to establish in the general case.
⃗ is not uniform in the coil width and is altered by the presence of the
Indeed, the applied magnetic field 𝐻
particles. For a sufficiently large applied magnetic field, the length scale of the magnetic field gradient is the
microparticle diameter, such that the integral can be considered constant over the volume of the microparticle.
Literature often introduces a simplified form of this equation using an ideal magnetic point dipole approximation
⃗ = 𝜇0 𝐻
⃗ , so the
and the overall response of the microparticle plus water system can be approximated by 𝐵
Equation 4 becomes:
⃗⃗⃗⃗
𝐹𝑚 ≈ 𝑉𝑝
∆𝜒
⃗ . ∇)𝐵
⃗ = 𝑉𝑝 𝑀∇(𝐵
⃗)
(𝐵
𝜇0
2 Experimental set up
Fig. S1 Top and side views of the experimental setup. The fluidic device is inserted in the center of an
electromagnetic coil (inner diameter of 5 cm) generating a maximum field intensity of 45 mT. This configuration
was chosen to provide a uniform magnetic field in the device. While the orientation of the field remains
perpendicular to the device, the field intensity can be tuned by controlling the current intensity in the coil (from 0
to 3 A). A USB camera was used to track the droplet trajectories. The flow in the device is controlled by an
external syringe pump.
3 Magnetic flux density mapping simulations in the vicinity of the rail
Fig. S2 (A) B vectors magnitude along y axis. The positive and negative values indicate that B vectors are
respectively oriented to the right or to the left along y axis. The dotted lines represent the edges of the magnetic
rail. (B) Mapping of B values in the (y, z) plane in the vicinity of the rail (cross section). The magnetic field lines
(black lines) are concentrated through the rail.
4 Flow profile on (x,y) plane
Fig. S3 Flow velocity magnitude along the y axis (perpendicular to the main flow) for a flow rate of 10 µL/s.
5 Droplet stability
An important aspect to be considered is the possibility of an eventual extraction of the particles from the
droplets. The magnetic particles cluster exerts an interfacial force on the droplet surface that can be defined as:
𝐹𝑐 = 𝛾 (
6𝜋2 𝑄
𝜙𝜌
1/3
)
(5)
Where 𝜙 is the compactness of magnetic beads, Q the quantity of magnetic beads (g), ρ the magnetic beads
density (g/m3) and 𝛾 the interface tension between the oil and the aqueous solution (N/m). In our case, 𝛾
between water and FC 40 oil (with 2% surfactant) was estimated around 10 mN.m-1. More detailed calculations
concerning Fc were previously published by our group (Ali-Cherif et al. 2012).
In our work, Fc has been evaluated at 10 µN, which is higher than the maximum value of Fm evaluated at about 4
µN. As observed in our experiments, Fc holds the magnetic beads within the droplet and extraction cannot
happen within the range of magnetic and drag forces values investigated here.
6 Droplet tracing simulations
Fig. S4 Simulation results showing the trajectory of droplets in a microchamber containing mechanical traps,
without any magnetic field applied for a flow rate of 5µL/s. The fives vertical black lines represent the inlet point
of the droplets in the simulation. The hydrodynamic resistance induced by the trapping structures modifies the
flow streamlines distribution. As a consequence, the droplets are spontaneously dragged around the trapping
structures and transported toward the outlet of the chamber (A). However, when the initial position of the droplet
is defined at the entrance of the trap (corresponding to the situation obtained through magnetic guiding), the
hydrodynamic drag forces drive the droplet within the trap and induce its confinement.
Supporting Videos
Video S1: Droplet motion in absence of external magnetic field. The droplet (400µm diameter containing 1.5 µg
of magnetic particles) was dragged with a flow rate of 5 µL/s. The droplet is moving from the inlet to the outlet
without noticeable influence of the rails.
Video S2: Droplet motion when applying an external magnetic field. The droplet (400µm diameter containing
1.5 µg of magnetic particles) was dragged at a flow rate of 5 µL/s. The magnetic field was turned ON when the
droplet trajectory intercepted the rail n°1. The droplet was stored in trap n°1.
Video S3: Droplet motion in presence of external magnetic field. The droplet (400µm diameter, 1.5 µg of
magnetic particles content) was dragged at a flow rate of 5 µL/s. The magnetic field was turned ON when
trajectory intercepted the rail n°4. The droplet was stored in trap n°4.
Video S4: Merging of two colored droplets by applying a flow rate of 15 µL/s and sequentially turning ON and
OFF the magnetic field to improve the mixing.
References
Ali-Cherif A, Begolo S, Descroix S, Viovy JL, Malaquin L (2012) Programmable Magnetic Tweezers and
Droplet Microfluidic Device for High-Throughput Nanoliter Multi-Step Assays. Angew Chem Int Ed
51(43):10765–10769