**2. Centrifugal hydrodynamics**

The rotationally controlled microfluidic "Lab-on-a-Disc" platform is based on (the volume density of) the centrifugal force

$$f\_{\alpha} = \rho r \,\alpha^2\tag{1}$$

the Euler force

$$f\_E = \rho r.d.o \,/\,\text{dt} \tag{2}$$

Lumped-Element Modeling for Rapid Design and Simulation of Digital Centrifugal Microfluidic Systems http://dx.doi.org/10.5772/62836 59

and the Coriolis force [1]

**1. Introduction**

58 Lab-on-a-Chip Fabrication and Application

infrastructures.

platform.

developed (Section 4).

the Euler force

are composed of a library of functional units.

**2. Centrifugal hydrodynamics**

density of) the centrifugal force

The centrifugal microfluidic platform has evolved into mature technology platform which has already proven to open significant market opportunity [1–5]. A large number of groups groups working on such LoaD systems in industry has already convincingly demonstrated the capability to integrate, automate, parallelize, and miniaturize a wide range of common bioanalytical test formats for detecting targets such as small molecules, proteins/antibodies, nucleic acids, and cells. Applications span from decentralized biomedical point-of-care diagnostics, veterinary medicine and agrifood, to the surveillance of the environment and

Based on the recently introduced, event-triggered flow control scheme [6, 7], highly functional microfluidic circuits can be assembled in a modular fashion from a limited set of LUOs to implement a broad repertoire of multi-step, multi-reagent bioassay protocols in a sample-toanswer fashion. Furthermore, it has been demonstrated that the chips could be progressively miniaturized to significantly enhance integration density (i.e., the number of assay steps and/ or tests per disc) and thus boost the overall cost efficiency and functionality of the LoaD

Similar to integrated circuits in microelectronics, a microfluidic network can be modeled by lumped-element descriptors. Instead of a finely meshed 3-dimensional lattice, the Lab-on-a-Disc systems are described by a low number of parameters such as pressure head (voltage source), flow resistance (electric resistance), and compressibility (capacitance). This reduceddimension method can be utilized for fast design and simulation of microfluidic systems that

After the introduction (Section 1), the hydrodynamic principles of centrifugal microfluidics are presented (Section 2) before outlining digital flow control schemes (Section 3). Next, the concept of lumped-element simulation in event-triggered centrifugal microfluidic networks is

The rotationally controlled microfluidic "Lab-on-a-Disc" platform is based on (the volume

(1)

(2)

<sup>2</sup> *f r* w = r w

. / *Ef r d dt* = r w

$$f\_{\mathbb{C}} = 2\,\mathsf{pcov} \tag{3}$$

where *ρ* is a fluid density on a rotating platform, *ω* = 2*πv* the angular velocity with the frequency of rotation *v*, *r* is a distance from a central axis, and *v* represents the speed of flow. All forces act in the plane of the disc and scale with the angular velocity *ω* directly impacts these three forces.

The centrifugal force (Eq. (1)) translates into an equivalent centrifugal pressure

$$
\Delta p\_{\alpha} = \rho \Delta r \overline{r} \,\alpha^2 \tag{4}
$$

and an average flow velocity

$$V = \frac{D\_h^{\text{-}} \rho \Delta r \phi^{\text{-}} \overline{r}}{32 \mu L} \tag{5}$$

of the liquid in the channel [5] featuring the hydraulic diameter *Dh* = 4*A*/*P* with the *A*, *P*, and *L*, its cross-sectional area *A*, wetted perimeter of the channel *P*, and length *L*. The fluid viscosity is denoted by *μ*, the mean radial position by *r*¯ =(*r*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*1)/ 2, and the radial length by ∆*r* = (*r*<sup>2</sup> − *r*1).

Air pockets, which often arise accidentally or strategically during priming, can be compressed by the hydrostatic pressure head of a liquid column (4) in a more central position. This centrifugally induced pressure compresses the enclosed gas volume

$$P\_c = P\_0 \frac{1}{1 - \Delta V / V} \tag{6}$$

according to Boyle's law [6] where *Pc* represents the pressure of the gas in the pneumatic chamber, *V* is the total volume of the pneumatic chamber, and ∆*V* denotes a reduction of gas volume due to filling of liquid in the pneumatic chamber.

#### **3. Digital flow control schemes**

Flow control is instrumental for orchestrating sequential liquid handling on the LoaD plat‐ forms where all volumes are subjected to the same centrifugal field (Eq. (1)). Such flow control can be categorized into rotationally actuated and instrument-supported schemes.

Instrument-supported valves involve some stationary modules (other than the platform innate spindle motor). To switch a valve, these "lab-frame" elements interact with the disc cartridge, either at rest or during spinning. The actuation can be powered by pneumatic pressure sources [8–9], heating of phase-change materials [10–13], or even varying the chip orientation with respect to the radial direction [14–16]. While these may provide enhanced and more flexible control, these active valving mechanisms typically involve additional instrumentation, maintenance, cost, and susceptibility to failure.

Rotationally actuated valves are far more common and are considered more suitable for deployment of inexpensive point-of-use applications. Through varying the rotationally induced fields relative to the statically defined forces such as interfacial or membrane tension, the force equilibrium at a fluid element can be unbalanced. Such static forces can be imple‐ mented by capillary action [17–21], dissolvable films (DFs) [22- 23], burstable foils [24], elastomeric membranes [25], dead-end pneumatic chambers [26], siphons [27]–29], and pneumatically enhanced centrifugo-pneumatic siphons (CPSVs) [30–33].

In particular the popular, rotationally actuated capillary "burst" valves are strongly dependent on physicochemical properties such as geometry, surface roughness, and contact angle; hence, valve performance is intimately linked to manufacturing fidelity. The often rather poor reproducibility and stability of these effects translate into a significant "smearing" of the burst frequencies. For serial flow control which is common in bioanalytical protocols, rather wide, non-overlapping bands of the spin rate have to be reserved for each assay step. As the maximum spin rate is practically limited by the motor power and safety, this imposes a practical limit on the number of sequential LUOs which can be rotationally controlled by a spindle motor.

Event-triggered valving circumvents this restriction [6– 7]. Here, the arrival of liquid at defined locations on the disc coordinates a sequential opening of valves; valve actuation is thus decoupled from changes in the spin rate and support instrumentation. So far, event-triggered valving has been based on dissolvable film (DF) membranes [22], [23], [34], [35] and, in function, can be described akin to an electrical relay. The architecture of the disc determines the order of valve actuation, while the timing is controlled by the dissolution of these mem‐ branes. It has been shown that event-triggered schemes can also implement logical flow control elements such as AND and OR conditions [6], thus enabling a modular system design similar to electronics. Developing the lumped-element tool for the simulation, digital centrifugal microfluidic systems can generate a broad scope of applications, thus mitigating development risks, upfront investment, and time to market.

The basic event-triggered valve is composed of a pneumatic chamber sealed by the restrained liquid and two dissolvable films called the load film (LF) and the control film (CF). The geometry of the pneumatic chamber is designed so that, at the spin rates typical for the centrifugal platform, the restrained liquid cannot be pumped into contact with the LF or CF by compressing the trapped air within the pneumatic chamber. Similarly, the section of the chamber connecting the LF and CF extends radially inward of the restrained liquid. When the CF is wetted and dissolved by an ancillary liquid, the pneumatic chamber is vented so the main liquid contacts and thus opens the LF.

However, the connecting channel between the LF and CF acts as a geometric barrier which prevents the liquid escaping through the disrupted CF. Thus, in this configuration, the CF acts analogous to the control line of an electrical relay and the LF to the load line. This basic configuration can then be arranged into a complex fluid network where valves sequentially cascaded; the flow released from the first valve triggers the subsequent "ancillary liquid." Importantly, the interval between valve actuations is governed by the aggregate time of membrane dissolution and liquid transfer.

either at rest or during spinning. The actuation can be powered by pneumatic pressure sources [8–9], heating of phase-change materials [10–13], or even varying the chip orientation with respect to the radial direction [14–16]. While these may provide enhanced and more flexible control, these active valving mechanisms typically involve additional instrumentation,

Rotationally actuated valves are far more common and are considered more suitable for deployment of inexpensive point-of-use applications. Through varying the rotationally induced fields relative to the statically defined forces such as interfacial or membrane tension, the force equilibrium at a fluid element can be unbalanced. Such static forces can be imple‐ mented by capillary action [17–21], dissolvable films (DFs) [22- 23], burstable foils [24], elastomeric membranes [25], dead-end pneumatic chambers [26], siphons [27]–29], and

In particular the popular, rotationally actuated capillary "burst" valves are strongly dependent on physicochemical properties such as geometry, surface roughness, and contact angle; hence, valve performance is intimately linked to manufacturing fidelity. The often rather poor reproducibility and stability of these effects translate into a significant "smearing" of the burst frequencies. For serial flow control which is common in bioanalytical protocols, rather wide, non-overlapping bands of the spin rate have to be reserved for each assay step. As the maximum spin rate is practically limited by the motor power and safety, this imposes a practical limit on the number of sequential LUOs which can be rotationally controlled by a

Event-triggered valving circumvents this restriction [6– 7]. Here, the arrival of liquid at defined locations on the disc coordinates a sequential opening of valves; valve actuation is thus decoupled from changes in the spin rate and support instrumentation. So far, event-triggered valving has been based on dissolvable film (DF) membranes [22], [23], [34], [35] and, in function, can be described akin to an electrical relay. The architecture of the disc determines the order of valve actuation, while the timing is controlled by the dissolution of these mem‐ branes. It has been shown that event-triggered schemes can also implement logical flow control elements such as AND and OR conditions [6], thus enabling a modular system design similar to electronics. Developing the lumped-element tool for the simulation, digital centrifugal microfluidic systems can generate a broad scope of applications, thus mitigating development

The basic event-triggered valve is composed of a pneumatic chamber sealed by the restrained liquid and two dissolvable films called the load film (LF) and the control film (CF). The geometry of the pneumatic chamber is designed so that, at the spin rates typical for the centrifugal platform, the restrained liquid cannot be pumped into contact with the LF or CF by compressing the trapped air within the pneumatic chamber. Similarly, the section of the chamber connecting the LF and CF extends radially inward of the restrained liquid. When the CF is wetted and dissolved by an ancillary liquid, the pneumatic chamber is vented so the

pneumatically enhanced centrifugo-pneumatic siphons (CPSVs) [30–33].

maintenance, cost, and susceptibility to failure.

60 Lab-on-a-Chip Fabrication and Application

risks, upfront investment, and time to market.

main liquid contacts and thus opens the LF.

spindle motor.

Alongside the basic configuration (**Figure 1**), the conditions of valve actuation can be altered by changing the arrangement of the CF. For example, locating the CF such that it can only be wetted when two or more upstream "ancillary liquid' volumes have been released (Figure 1) establishes a Boolean AND condition. Similarly, by designing a valve with two CFs where wetting one or the other will trigger the valve, we create a Boolean OR condition (**Figure 2**). Finally, locating two CFs in close proximity so they can be reached by a single ancillary liquid can simultaneously open two pneumatically connected valves and thus can represent parallel valve actuation (**Figure 3**).

**Figure 1.** Schematic demonstrating the basic event-triggered configuration and also showing the Boolean AND release mechanism (a) Valve closed, (b) upstream valve 1 opens (c) AND upstream valve 2 opens, (d) CF is dissolved, (e) LF is wetted, (f) Valve opens.

**Figure 2.** Schematic illustrating the OR conditional release mechanism. The top pane shows the valve actuation trig‐ gered by liquid movement to one chamber and the lower pane shows the valve actuation triggered. (a) Valve closed, (b) CF wetted, (c) LF wetted, (d) Valve opens.

**Figure 3.** Schematic illustrating the OR conditional release mechanism. The top pane shows the valve actuation trig‐ gered by liquid movement to one chamber and the lower pane shows the valve actuation triggered. (a) Valve closed, (b) CF wetted, (c) LF wetted, (d) Valve opens.
