30
3 Simulating Droplet Microfluidic Networks
In the following, first, the basic working principles of the simulation framework
are provided. In Sect. 3.3.2, physical phenomena are considered which cannot be
simulated by previously proposed approaches. Section 3.3.3 describes how support
for these so far unsupported physical phenomena can easily be added to the proposed
framework. Afterwards, Sect. 3.3.4 summarizes the event-based algorithm. Finally,
in Sect. 3.3.5 the correct simulation of those physical phenomena is evaluated.
3.3.1 Basic Working Principle
This section introduces the basic working principle of the proposed simulation
framework for droplet microfluidic networks. The scheme applied here is similar to
the one applied in other simulations such as [4, 7, 21, 37, 40, 67, 101, 105, 106, 109].
As introduced in Definition 2.1 (cf. page 16), microfluidic networks are represented in terms of a directed graph. Using this graph, the simulation framework
automatically derives, applies, and solves microfluidic equations derived from the
1D analysis model as described in Sect. 3.2. This gives a flow state where also the
effects of droplets are considered. As already discussed, this flow state becomes
invalid as soon as a new droplet is injected, any droplet leaves the network, or
any droplet enters another edge. Consequently, the entire equation system has to
be frequently updated and re-calculated—which would be impossible by hand. The
developed simulator conducts this automatically. To this end, the simulator repeatedly performs the following steps, which are similar to that proposed in [101]:
1. Compute Flow State: The simulator computes the flow state (i.e., the pressures
and volumetric flow rates in all channels) by considering all droplets and their
current positions. Therefore, the simulator automatically derives an equation
system which considers the resistances caused by all droplets and solves it. The
resulting volumetric flow rates allow the determination of the droplet speeds,
i.e. by Eq. 3.7 from Sect. 3.2.
2. Compute Next Event Time: The next event time is defined by the minimum time
until a new droplet is injected, any droplet leaves the network, or any droplet
enters the next channel/module (here the droplets’ speeds are used).
3. Update System State: Finally, the simulator updates the equation system so that
it represents the state at this event time (i.e., updates the droplet positions). For
the droplet causing the event, the current flow state decides which channel this
droplet enters next (i.e., the channel with the highest volumetric flow rate).
By repeatedly performing these steps in the simulator, all interdependencies
caused by droplets are considered. Furthermore, these event-based calculations
make the algorithm efficient and allow to simulate large microfluidic networks
in negligible computation times. Overall, this provides the basic principle of an
efficient simulation of droplet microfluidic networks.
3 Simulating Droplet Microfluidic Networks
In the following, first, the basic working principles of the simulation framework
are provided. In Sect. 3.3.2, physical phenomena are considered which cannot be
simulated by previously proposed approaches. Section 3.3.3 describes how support
for these so far unsupported physical phenomena can easily be added to the proposed
framework. Afterwards, Sect. 3.3.4 summarizes the event-based algorithm. Finally,
in Sect. 3.3.5 the correct simulation of those physical phenomena is evaluated.
3.3.1 Basic Working Principle
This section introduces the basic working principle of the proposed simulation
framework for droplet microfluidic networks. The scheme applied here is similar to
the one applied in other simulations such as [4, 7, 21, 37, 40, 67, 101, 105, 106, 109].
As introduced in Definition 2.1 (cf. page 16), microfluidic networks are represented in terms of a directed graph. Using this graph, the simulation framework
automatically derives, applies, and solves microfluidic equations derived from the
1D analysis model as described in Sect. 3.2. This gives a flow state where also the
effects of droplets are considered. As already discussed, this flow state becomes
invalid as soon as a new droplet is injected, any droplet leaves the network, or
any droplet enters another edge. Consequently, the entire equation system has to
be frequently updated and re-calculated—which would be impossible by hand. The
developed simulator conducts this automatically. To this end, the simulator repeatedly performs the following steps, which are similar to that proposed in [101]:
1. Compute Flow State: The simulator computes the flow state (i.e., the pressures
and volumetric flow rates in all channels) by considering all droplets and their
current positions. Therefore, the simulator automatically derives an equation
system which considers the resistances caused by all droplets and solves it. The
resulting volumetric flow rates allow the determination of the droplet speeds,
i.e. by Eq. 3.7 from Sect. 3.2.
2. Compute Next Event Time: The next event time is defined by the minimum time
until a new droplet is injected, any droplet leaves the network, or any droplet
enters the next channel/module (here the droplets’ speeds are used).
3. Update System State: Finally, the simulator updates the equation system so that
it represents the state at this event time (i.e., updates the droplet positions). For
the droplet causing the event, the current flow state decides which channel this
droplet enters next (i.e., the channel with the highest volumetric flow rate).
By repeatedly performing these steps in the simulator, all interdependencies
caused by droplets are considered. Furthermore, these event-based calculations
make the algorithm efficient and allow to simulate large microfluidic networks
in negligible computation times. Overall, this provides the basic principle of an
efficient simulation of droplet microfluidic networks.