**4. Magnetic actuation in LOC: principles and exploited hardware**

Some applications of magnetic fields in LOCs have already been mentioned and range from magnetic separation for chemical and biological analyses to sample manipulation for drug screening and cell sorting. In terms of magnetic actuation of samples, it is possible to classify the various applications in two main categories: (1) magnetic separation and (2) magnetic manipulation. In the first case, two or more classes of objects are separated depending on their magnetic properties, but without any need to properly drive them along complex paths or to guarantee the execution of specific tasks; in this case, magnetic fields are responsible for separation, but transportation is usually provided by fluidic forces. In the second case, a more accurate control is required to enable a single magnetic object or a swarm of them to follow a planned trajectory or to perform a specific task; larger magnetic fields and forces are required in this case, as they are responsible also for object transport.

#### **4.1. Magnetic separation**

Magnetic separation, often defined as magnetophoresis, is widely exploited in LOC applica‐ tions. Magnetophoresis is a nondestructive method for the selective collection or separation of magnetic particles, by moving them in a viscous medium under the influence of an applied magnetic field [16]. Usually, in LOC applications, we refer to free-flow magnetophoresis, since the separation of particles or magnetic objects takes place in a liquid environment where magnetic particles are deflected from the direction of laminar flow by a perpendicular magnetic field (**Figure 3**); the extent of the deflection depends mainly on flow rate and on the susceptibility of the magnetic particle, or more precisely, on the susceptibility mismatch between the particle and the fluid.

The vector *udefl*, which represents the deflection of magnetic particles due to the applied magnetic field, is the result of two contributions: the flow velocity induced on the particle by the applied magnetic field umag, and the hydrodynamic flow velocity uhyd:

sensing coils; by measuring the resulting voltage induced into the coil, it is possible to obtain sample magnetic moment and to magnetically characterize it. The sensitivity of this kind of technique can reach 10−6 emu. When the samples are really diluted or show really weak magnetic properties, thus claiming for higher sensitivities, SQUID-based magnetometry can be a suitable solution, enabling to reach sensitivities up to 10−8–10−12 emu. The magnetic properties of the material are measured by detecting quantum mechanical effects in conjunc‐ tion with superconducting detection coils. In both VSM and SQUID magnetometry, however, the duration of a single measurement is in the order of some hours. This obviously represents a strong limitation for all cases in which the characterization of a large number of samples is needed. On the other hand, force-based methods, such as Gouy and Faraday balances, rely on the change in weight of a magnetic material when it is subjected to a uniform magnetic field. Commercial systems based on the Faraday method, such as the alternating gradient magne‐ tometer (AGM), provide sensitivities in the 10−8–10−9 emu range with really fast measurement

**4. Magnetic actuation in LOC: principles and exploited hardware**

in this case, as they are responsible also for object transport.

Some applications of magnetic fields in LOCs have already been mentioned and range from magnetic separation for chemical and biological analyses to sample manipulation for drug screening and cell sorting. In terms of magnetic actuation of samples, it is possible to classify the various applications in two main categories: (1) magnetic separation and (2) magnetic manipulation. In the first case, two or more classes of objects are separated depending on their magnetic properties, but without any need to properly drive them along complex paths or to guarantee the execution of specific tasks; in this case, magnetic fields are responsible for separation, but transportation is usually provided by fluidic forces. In the second case, a more accurate control is required to enable a single magnetic object or a swarm of them to follow a planned trajectory or to perform a specific task; larger magnetic fields and forces are required

Magnetic separation, often defined as magnetophoresis, is widely exploited in LOC applica‐ tions. Magnetophoresis is a nondestructive method for the selective collection or separation of magnetic particles, by moving them in a viscous medium under the influence of an applied magnetic field [16]. Usually, in LOC applications, we refer to free-flow magnetophoresis, since the separation of particles or magnetic objects takes place in a liquid environment where magnetic particles are deflected from the direction of laminar flow by a perpendicular magnetic field (**Figure 3**); the extent of the deflection depends mainly on flow rate and on the susceptibility of the magnetic particle, or more precisely, on the susceptibility mismatch

procedures [14, 15].

42 Lab-on-a-Chip Fabrication and Application

**4.1. Magnetic separation**

between the particle and the fluid.

$$
\mathfrak{u}\_{dgt} = \mathfrak{u}\_{mag} + \mathfrak{u}\_{hyd} \tag{9}
$$

The magnetically induced flow velocity umag, can be expressed as the ratio of the magnetic force Fm exerted on the particle to the viscous drag force:

$$
\mu\_{\text{mag}} = \frac{F\_m}{F\_d} = \frac{F\_w}{6\pi\eta r} \tag{10}
$$

In a magnetophoresis application, the magnetic force depends on the particle features, mainly its volume Vp, on the mismatch in terms of magnetic properties between the particle and the fluid, and on the applied magnetic field B. Eq. (5) becomes consequently [17]:

$$F\_{\kappa} = \frac{V\_{\rho} \cdot \Delta \mathcal{X}}{\mu\_0} (\mathbf{B} \cdot \nabla) \mathbf{B} \tag{11}$$

Eq. (11) is suitable for both paramagnetic and superparamagnetic particles, since soft mag‐ netism approximation, and lack of magnetic memory is considered for the particles, and for relatively high magnetic field strengths, able to induce in the particles a magnetization close to the saturation one. To this aim, macroscopic permanent magnets and electromagnets can be exploited, since they produce sufficiently strong fields (>0.5 T), able to saturate superpara‐ magnetic particles.

**Figure 3.** Typical schematization of a magnetophoresis scenario in which a magnetic field perpendicular to the flow direction is exploited to deviate magnetic particles from the trajectory imposed by fluidic forces and thus to separate them from the rest of the sample. Complete separation of the species can be obtained through some consecutive steps.

It is possible to distinguish between positive and negative magnetophoresis depending on the sign of the susceptibility mismatch: if Δχ is positive, for example, in the case of magnetic micro-/ nano-objects in a non-magnetic fluid, we talk about positive magnetophoresis and the particles are attracted by the externally applied magnetic field. On the other hand, when the suscepti‐ bility mismatch Δχ is negative, for example, in the case of diamagnetic particles in a magne‐ tofluid, the particles are repelled from the magnetic field and negative magnetophoresis occurs. When designing a magnetophoresis device, it is necessary to assure the dominance of magnetic forces respect to the other physical contributions. Considering Eq. (11), Fm results proportional to the magnetic field gradients and to the susceptibility mismatch. Several strategies aiming at optimizing magnetophoresis have been investigated and proposed, and they may be essentially classified in two categories distinguishing between those aiming at maximizing the magnetic field gradients and those acting on the susceptibility mismatch [18].

In the attempt to increase the magnetic field gradient, many solutions have been proposed in literature, all aiming at the generation of a nonuniform magnetic field distribution. In some cases, uniform external magnetic field sources, such as permanent magnets, were combined with wires [19], ferromagnetic strings [20], or magnetic microparticles embedded in the chip structure itself [21], whereas in other cases, the integration of small electromagnets produced intense magnetic field gradients [22]. Alternatively, permanent magnets can be actuated to generate a time-dependent magnetic field, for example, through the use of a rotating magnetic wheel [23], or arranged in an asymmetric configuration, thus generating spatial field variation or multiple magnetic separation stages [24]. One of the most commonly employed magnet configuration in magnetophoresis applications, able to maximize field distribution anisotropy, is the quadrupolar arrangement which creates a magnetic gradient radially outward from the center of the flow column [25]. In a variant of this, the fluid is static, while an applied magnetic field is moved up the container [26]. The particles move in the resulting field gradient at a velocity dependent on their magnetophoretic mobility. At the top of the container, they enter a removable section and are held here by a permanent magnet. The bottom section of the container moves to the next section where a magnetic field with different strength to the first is applied and the process repeats. The result is a fractionation of the sample into aliquots differing for magnetophoretic mobility [27].

The alternative strategy to enhance the magnetic separation capabilities of the device lies in increasing the susceptibility mismatch by modifying either the susceptibility of the particle, or the one of the surrounding fluid. This can be accomplished (1) by labeling the cells or the desired microstructure with higher magnetic susceptibility beads (2) by internalizing higher quantity of magnetic material, and (3) by using a ferrofluid medium, for example, by adding gadolinium, and diamagnetic particles instead of the para/ferromagnetic ones [28].

#### **4.2. Magnetic micromanipulation**

Magnetic fields can be employed in LOC for the non-contact manipulation of biological samples or other magnetically labeled substances/structures. Apart from applications in which it is necessary to separate different types of samples or specific entities from the medium (tasks that can benefit from techniques mainly based on magnetophoresis), in some cases, precise manipulation or transport along defined paths is required. This kind of task is more complex in terms of extent of magnetic fields required and of control, considering that for 3D manip‐ ulation, torque equilibrium must be taken into account, in addition to force balance.

nano-objects in a non-magnetic fluid, we talk about positive magnetophoresis and the particles are attracted by the externally applied magnetic field. On the other hand, when the suscepti‐ bility mismatch Δχ is negative, for example, in the case of diamagnetic particles in a magne‐ tofluid, the particles are repelled from the magnetic field and negative magnetophoresis occurs. When designing a magnetophoresis device, it is necessary to assure the dominance of magnetic forces respect to the other physical contributions. Considering Eq. (11), Fm results proportional to the magnetic field gradients and to the susceptibility mismatch. Several strategies aiming at optimizing magnetophoresis have been investigated and proposed, and they may be essentially classified in two categories distinguishing between those aiming at maximizing the magnetic field gradients and those acting on the susceptibility mismatch [18].

In the attempt to increase the magnetic field gradient, many solutions have been proposed in literature, all aiming at the generation of a nonuniform magnetic field distribution. In some cases, uniform external magnetic field sources, such as permanent magnets, were combined with wires [19], ferromagnetic strings [20], or magnetic microparticles embedded in the chip structure itself [21], whereas in other cases, the integration of small electromagnets produced intense magnetic field gradients [22]. Alternatively, permanent magnets can be actuated to generate a time-dependent magnetic field, for example, through the use of a rotating magnetic wheel [23], or arranged in an asymmetric configuration, thus generating spatial field variation or multiple magnetic separation stages [24]. One of the most commonly employed magnet configuration in magnetophoresis applications, able to maximize field distribution anisotropy, is the quadrupolar arrangement which creates a magnetic gradient radially outward from the center of the flow column [25]. In a variant of this, the fluid is static, while an applied magnetic field is moved up the container [26]. The particles move in the resulting field gradient at a velocity dependent on their magnetophoretic mobility. At the top of the container, they enter a removable section and are held here by a permanent magnet. The bottom section of the container moves to the next section where a magnetic field with different strength to the first is applied and the process repeats. The result is a fractionation of the sample into aliquots

The alternative strategy to enhance the magnetic separation capabilities of the device lies in increasing the susceptibility mismatch by modifying either the susceptibility of the particle, or the one of the surrounding fluid. This can be accomplished (1) by labeling the cells or the desired microstructure with higher magnetic susceptibility beads (2) by internalizing higher quantity of magnetic material, and (3) by using a ferrofluid medium, for example, by adding

Magnetic fields can be employed in LOC for the non-contact manipulation of biological samples or other magnetically labeled substances/structures. Apart from applications in which it is necessary to separate different types of samples or specific entities from the medium (tasks that can benefit from techniques mainly based on magnetophoresis), in some cases, precise manipulation or transport along defined paths is required. This kind of task is more complex

gadolinium, and diamagnetic particles instead of the para/ferromagnetic ones [28].

differing for magnetophoretic mobility [27].

44 Lab-on-a-Chip Fabrication and Application

**4.2. Magnetic micromanipulation**

It has been demonstrated that biological systems or chemical samples labeled with magnetic nanoparticles can be micro-/nano-manipulated or transported in three dimensions, by exploiting combinations of electromagnets or permanent magnets, possibly moved or rotated along three axes. Permanent magnets offer the advantage to produce large fields without the need of any electrical current, thus avoiding powering, heating and control issues, which have to be faced instead when using electromagnets. On the other hand, electromagnets offer the possibility to tune magnetic field gradients and field intensities by simply varying the currents across the coils. By properly combining electromagnets, it is possible to produce in the workspace both varying magnetic fields, without the need of moving parts, and spatially uniform magnetic fields and gradients. This makes possible to accomplish also quite complex manipulation and locomotion paradigms. Several architectures have been proposed, present‐ ing different magnet and electromagnet arrangements. In LOC applications, due to the need to finely control the locomotion of small-scale entities, electromagnets are the most commonly employed solution.

Generally, when an electrical current flows in a wire, a magnetic field is generated according to the Biot–Savart theory [9]. When considering a single circular coil in which a current with magniture *I* is flowing, the magnetic field along the central axis of the coil can be defined as:

$$\mathbf{B} = \begin{array}{c} d\mathbf{B} = \frac{I r^2}{2 \left(r^2 + z^2\right)^{\frac{3}{2}}}\\ 2 \left(r^2 + z^2\right)^{\frac{3}{2}} \end{array} \tag{12}$$

where *r* is the radius of the coil, and *z* is the coordinate along the central axis.

Often, specific coil pairs arrangements, namely Helmholtz and Maxwell coils, are exploited in micromanipulation applications. They consist of two identical circular magnetic coils sym‐ metrically placed along a common axis, one on each side of the workspace, and separated by a distance *d* corresponding to coil radius (rH) in the case of Helmhotz coils and to √3rM in the case of Maxwell coils. In Helmholtz arrangement, each coil carries an equal electric current in the same direction, whereas in Maxwell coils currents flow in opposite directions. The magnetic flux density in case of Helmholtz and Maxwell coils can be derived from Eq. (12):

$$\mathcal{B}\_{H} = \frac{\mu\_{0}r\_{H}^{2}N\_{H}\mathbf{I}\_{H}}{2} \left( \frac{1}{\left[r^{2} + \left(\frac{d}{2} - z\right)^{2}\right]^{\frac{3}{2}}} + \frac{1}{\left[r^{2} + \left(\frac{d}{2} + z\right)^{2}\right]^{\frac{3}{2}}} \right) \tag{13}$$

$$\mathbf{B}\_{M} = \frac{\mu\_{0}r\_{M}^{2}N\_{M}\mathbf{I}\_{M}}{2} \left( \frac{1}{\left[r^{2} + \left(\frac{d}{2} - z\right)^{2}\right]^{\frac{3}{2}}} - \frac{1}{\left[r^{2} + \left(\frac{d}{2} + z\right)^{2}\right]^{\frac{3}{2}}} \right) \tag{14}$$

NH,IH,rH, NM,IM,rM are the numbers of windings, current, and radius of Helmholtz and Maxwell coils, respectively.

When considering a combination of Helmholtz and Maxwell coils (**Figure 4A**), the magnetic field B and magnetic field gradient ∇B in the workspace can be derived analytically by Eqs. (13) and (14) as follows:

$$\mathbf{B} = \frac{8\mu\_0 N\_H I\_H}{5\sqrt{5}r\_H} \tag{15}$$

$$\mathbf{B} = \frac{48\sqrt{3}\mu\_0 N\_M I\_M}{49\sqrt{7}r\_M^2} \tag{16}$$

Equations (15) and (16) clearly show that Helmholtz coils are able to generate a uniform magnetic field, whereas Maxwell coils produce a uniform magnetic field gradient along its axis. For this reason, combinations of Helmholtz and Maxwell coils have been exploited to obtain both a uniform field gradient and magnetic field uniformity [29].

Nonuniform field setups have been developed as well. Despite the major complexity both in terms of design/fabrication and control, they enable an increase in the number of controlled degrees of freedom. In this sense, a representative example is the OctoMag system [30, 31] (**Figure 4B**), designed for the control of intraocular microrobots for minimally invasive retinal therapy and diagnosis, but showing also potentialities for use as a wireless micromanipulation apparatus. It consists of eight stationary electromagnets with soft-magnetic cores able to generate predefined values of magnetic field and gradient, providing the manipulated object with five degrees of freedom; this system can operate closed-loop position control by exploiting computer-assisted visual tracking or in open loop by relying only on the operator microscopemediated visual feedback. Alternative approaches aim at exploiting other sources of nonuni‐ form magnetic fields: Martel et al. [32] demonstrated the effectiveness of using an MRI scanner for the control of a swarm of magnetotactic bacteria in executing a manipulation task on microobjects. Micro-assembly of micro-objects using a cluster of microparticles (with average diameter of 100 μm) and a magnetic-based manipulation system has also been shown in [33].

**Figure 4.** Different magnetic field generation setup exploited for magnetic manipulation. (A) Combination of Helm‐ holtz and Maxwell coils; (B) OctoMag system [31].
