**1. Introduction**

The need of reliable, precise, and fast techniques for biochemical and biological analysis has fostered the search for miniaturized systems integrating multiple laboratory techniques, assays, and procedures into a really small chip, up to few square centimeters in size. These small platforms, named lab-on-a-chip (LOC) or, less frequently, micro total analysis systems (μTAS), have historically been fabricated in silicon and/or glass using semiconductor processing techniques. More recently, polymer-based devices emerged, thanks to the introduction of soft

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lithography [1]. LOC devices can be considered as sophisticate microsystems embedding mechanical, electronic, and fluidic functions [2], aiming at mixing, pumping, and manipulat‐ ing samples. It is possible to identify a wide literature concerning LOC systems, in which a variety of applications ranging from biological assays, drug sorting and testing, DNA extrac‐ tion, cell manipulation, etc., have been explored.

The use of LOC devices for laboratory tasks execution offers several advantages: reduced sample and reagent volumes, fast sample processing, high sensitivity and spatial resolution, increased detection accuracy, low contamination, high throughput, and reliability thanks to the possibility to automate some processes, without depending on human operator skills [3].

Due to the really small dimensions of LOC devices, the major role is played by surface effects with respect to inertial ones. Consequently, traditional actuation strategies cannot be exploited for actuation in LOCs. Fluidic actuation is the most commonly employed strategy, but electrostatic, magnetic, and chemical motion has been reported as well.

LOC systems can be distinguished in continuous-flow and stationary devices, depending on the role played by the fluidic actuation in the execution of the desired tasks. In continuousflow devices, microfluidic forces are responsible for the effects experienced by the sample (e.g., beads, liquids, or droplets). In static flow devices, although the working environment is still a fluid, additional actuation strategies, such those based on magnetic fields, are exploited for effectively executing the desired task.

Depending on the working environment and on the object lengthscale, the most effective physical principle to be exploited in order to achieve the desired objective can change signif‐ icantly. **Figure 1** shows the trend of different physical effects at varying of the manipulated object dimension. At the microscale, due to the capillary forces and to low Reynolds numbers, it is quite hard to manipulate or mix liquids and particles by exploiting only fluidic forces or direct manipulation, and the exploitation of other actuation strategies showing high efficiency at the microscale is required. In this sense, magnetic field-based strategies exploitation could be a valid solution. In LOC scenarios, in fact, the magnetic field sources can be really close to the working environment, thus compensating the rapid decay of magnetic force with the distance between the source and the object [4]. Furthermore, the exploitation of magnetic fields enables non-contact manipulation [5] also for biological samples, thus paving the way for a wide variety of applications in biology and medicine. In this chapter, the force balance acting on a micro-object in a LOC will be analyzed with a particular focus on magnetism basic theory. The exploitation of magnetic fields for torque and force generation will be considered, especially for magnetic separation and magnetic manipulation applications. Techniques employed both to endow an object with magnetic properties and to characterize it will be described. Finally, potential applications of magnetic field-based strategies in LOCs will be reviewed. Throughout the chapter, technologies and examples not typical of LOCs but deriving, for example, from the world of microrobotics will be introduced, thus foreseeing a deeper and deeper interaction/integration between these two fields.

**Figure 1.** Scaling of different forces in function of the size of the object.

#### **2. Physics at the microscale: key principles**

According to Newton's second law, when considering a magnetic microcarrier with mass *mp* and moving in a fluidic environment with velocity *v*, the following equilibrium equation should be considered:

$$m\_{\rho} \frac{d\mathbf{v}}{dt} = \mathbf{F}\_m + \mathbf{F}\_g + \mathbf{F}\_d + \mathbf{F}\_b \tag{1}$$

Several physical effects, including the magnetic force *Fm*, the fluidic drag force *Fd*, the net gravitational force *Fg* that take into account also the fluidic lift effect, and the Brownian interaction force *Fb* contribute to the force balance of the moving object. In the following, the Brownian force, representing fluid-object and inter-objects interactions, will be neglected as it is really weak with respect to the other contributions. The other forces contributing to the equilibrium will be analyzed more in detail.

#### **2.1. Fluidic drag force**

lithography [1]. LOC devices can be considered as sophisticate microsystems embedding mechanical, electronic, and fluidic functions [2], aiming at mixing, pumping, and manipulat‐ ing samples. It is possible to identify a wide literature concerning LOC systems, in which a variety of applications ranging from biological assays, drug sorting and testing, DNA extrac‐

The use of LOC devices for laboratory tasks execution offers several advantages: reduced sample and reagent volumes, fast sample processing, high sensitivity and spatial resolution, increased detection accuracy, low contamination, high throughput, and reliability thanks to the possibility to automate some processes, without depending on human operator skills [3].

Due to the really small dimensions of LOC devices, the major role is played by surface effects with respect to inertial ones. Consequently, traditional actuation strategies cannot be exploited for actuation in LOCs. Fluidic actuation is the most commonly employed strategy, but

LOC systems can be distinguished in continuous-flow and stationary devices, depending on the role played by the fluidic actuation in the execution of the desired tasks. In continuousflow devices, microfluidic forces are responsible for the effects experienced by the sample (e.g., beads, liquids, or droplets). In static flow devices, although the working environment is still a fluid, additional actuation strategies, such those based on magnetic fields, are exploited for

Depending on the working environment and on the object lengthscale, the most effective physical principle to be exploited in order to achieve the desired objective can change signif‐ icantly. **Figure 1** shows the trend of different physical effects at varying of the manipulated object dimension. At the microscale, due to the capillary forces and to low Reynolds numbers, it is quite hard to manipulate or mix liquids and particles by exploiting only fluidic forces or direct manipulation, and the exploitation of other actuation strategies showing high efficiency at the microscale is required. In this sense, magnetic field-based strategies exploitation could be a valid solution. In LOC scenarios, in fact, the magnetic field sources can be really close to the working environment, thus compensating the rapid decay of magnetic force with the distance between the source and the object [4]. Furthermore, the exploitation of magnetic fields enables non-contact manipulation [5] also for biological samples, thus paving the way for a wide variety of applications in biology and medicine. In this chapter, the force balance acting on a micro-object in a LOC will be analyzed with a particular focus on magnetism basic theory. The exploitation of magnetic fields for torque and force generation will be considered, especially for magnetic separation and magnetic manipulation applications. Techniques employed both to endow an object with magnetic properties and to characterize it will be described. Finally, potential applications of magnetic field-based strategies in LOCs will be reviewed. Throughout the chapter, technologies and examples not typical of LOCs but deriving, for example, from the world of microrobotics will be introduced, thus foreseeing a

electrostatic, magnetic, and chemical motion has been reported as well.

deeper and deeper interaction/integration between these two fields.

tion, cell manipulation, etc., have been explored.

36 Lab-on-a-Chip Fabrication and Application

effectively executing the desired task.

Navier–Stokes equations completely define a fluid velocity in space and time. From these equations, it is possible to derive the Reynolds number (*Re*), a dimensionless quantity that is proportional to the ratio between the fluid's inertia and its viscosity and that allows to define a fluid's behavior when it flows around an object. Given the fluid density *ρ*, the dynamic viscosity *η*, the object maximum velocity with respect to the fluid *v* and a characteristic linear dimension *L, Re* can be defined as:

$$Re = \frac{\rho \nu L}{\eta} \tag{2}$$

Usually, in both microrobotics and LOC applications, a low-*Re* regime, typically defined for *Re* < 10, applies. At low *Re*, surface and capillary forces play an important role compared to inertia and temporal variations of the flow pattern. Due to the relative importance of surface effects, flow at low *Re*-number strongly depends on object geometry. Thus, it is interesting to derive the viscous drag force acting on the object. By approximating the object to a sphere with radius *r* put in an infinite extent of fluid, the viscous drag force can be calculated as a linear function of the sphere's velocity through the fluid:

$$F\_d = 6\pi\eta r(\Psi\_f - \Psi\_p) \tag{3}$$

where *vf* and *vp* are the fluid and the sphere velocity, respectively.

#### **2.2. Gravitational force**

Inertial and gravitational forces play a minor role at low *Re*. When considering a micro-object immersed in a liquid, usually net gravitational force is taken into account. In fact upthrust forces, responsible for body buoyancy, should be considered in addition to gravitational force, which acts in the opposite direction:

$$F\_g = -V\_p \left(\rho\_p - \rho\_f\right) \mathbf{g} \tag{4}$$

In Eq. (4), *Vp* and *ρ<sup>p</sup>* are object volume and density, *ρ*<sup>f</sup> is the fluid density, and *g* is the gravity acceleration.

#### **2.3. Magnetic force**

The force acting on an object immersed in a magnetic field depends both on the field features and on the object properties. The magnetic force acting on a magnetic microstructure can be modeled by using the "effective" dipole moment method, in which a magnetic object is replaced by an "equivalent" point dipole with a moment mp,eff [6]. The force on the dipole is given by:

$$F\_w = \mu\_f \left(\mathfrak{m}\_{p, \text{gf}} \cdot \nabla\right) \mathbf{B} \tag{5}$$

where μf is the magnetic permeability of the medium, mp,eff is the "effective" dipole moment of the object, and B is the magnetic field produced by an external source acting at the center of the target object, where the equivalent point dipole is located.

The dipole moment *m* strictly depends on object volume and magnetic properties and it can be defined as:

$$\mathbf{m} = \mathbf{M} \cdot V \tag{6}$$

where M and V are the magnetization and the volume of the dipole, respectively.

viscosity *η*, the object maximum velocity with respect to the fluid *v* and a characteristic linear

*vL Re* r

6( ) *d fp F* = ph*r* n

and *vp* are the fluid and the sphere velocity, respectively.

n

Inertial and gravitational forces play a minor role at low *Re*. When considering a micro-object immersed in a liquid, usually net gravitational force is taken into account. In fact upthrust forces, responsible for body buoyancy, should be considered in addition to gravitational force,

In Eq. (4), *Vp* and *ρ<sup>p</sup>* are object volume and density, *ρ*<sup>f</sup> is the fluid density, and *g* is the gravity

The force acting on an object immersed in a magnetic field depends both on the field features and on the object properties. The magnetic force acting on a magnetic microstructure can be modeled by using the "effective" dipole moment method, in which a magnetic object is replaced by an "equivalent" point dipole with a moment mp,eff [6]. The force on the dipole is

*F g g pp f* =- - *V* (r r

h

Usually, in both microrobotics and LOC applications, a low-*Re* regime, typically defined for *Re* < 10, applies. At low *Re*, surface and capillary forces play an important role compared to inertia and temporal variations of the flow pattern. Due to the relative importance of surface effects, flow at low *Re*-number strongly depends on object geometry. Thus, it is interesting to derive the viscous drag force acting on the object. By approximating the object to a sphere with radius *r* put in an infinite extent of fluid, the viscous drag force can be calculated as a linear

<sup>=</sup> (2)

(3)

) (4)

, ( ) *FmB m f p eff* = ×Ñ *µ* (5)

dimension *L, Re* can be defined as:

38 Lab-on-a-Chip Fabrication and Application

where *vf*

acceleration.

given by:

**2.3. Magnetic force**

**2.2. Gravitational force**

which acts in the opposite direction:

function of the sphere's velocity through the fluid:

As mentioned, the force exerted on such dipole varies upon the features of the magnetic field sources. It also depends on the distance between the source and the target object. If considering a permanent magnet, the magnetic field density at a generic point P can be expressed as:

$$\mathbf{B} = \frac{\mu\_0}{4\pi} \left( \frac{\Im(\mathbf{m} \cdot \mathbf{r}) \, r}{\left| \, \left| \, \mathbf{r} \right|^{\mathcal{S}}} - \frac{\mathbf{m}}{\left| \, \left| \, \mathbf{r} \right|^{\mathcal{S}}} \right) \right) \tag{7}$$

with *r* being the distance vector connecting the field source and the point P.
