8.1 Discrete Model
101
and, afterwards, it is shown how an instance of the model for a given microfluidic
network can be derived (Sect. 8.1.2). An evaluation of the precision of the model is
provided in Sect. 8.1.3.
8.1.1 Definition of the Discrete Model
The proposed discrete model is based on the following main concepts: (1) a discrete
representation of time, (2) a distinction between payload and header droplets,
(3) a discrete consideration of droplet behavior at bifurcations and sorters, and
(4) constraints restricting the distance of droplets to avoid coalescences of droplets.
These concepts are briefly discussed next before the resulting model is illustrated
using an example.
Discrete Representation of Time: The model discretizes the continuous time during
the droplet flow into atomic time steps. This allows to describe the duration a droplet
requires to flow through a channel or to execute a module’s operation in terms of a
number of time steps.
Distinction between Payload/Header Droplets: Since the payload droplets and the
header droplets can be of different volumes and fluids, they can cause different resistances in the channels and modules. Accordingly, the discrete model differentiates
between these droplet types. More precisely, the number of time steps a payload
droplet requires to flow through a channel c ∈ C or to execute a module m ∈ M
is defined by the function pSteps : C ∪ M → N. Accordingly, the function
hSteps : C → N defines the respective number of time steps for header droplets
to flow through a channel c ∈ C. Hence, depending on the type, a droplet takes a
certain amount of time steps before it enters the succeeding channel or module in
the network.
Behavior at Bifurcations/Sorters: The physical behavior at bifurcations for passive
droplet routing is abstracted by taking the number of time steps into account which
is required for the respective droplet to flow through either of the successor channels.
More precisely, a droplet flows into the successor channel requiring the least amount
of time steps. If this channel already contains a droplet, it flows into the other
successor channel. This behavior is accordingly applied if both channels already
contain a droplet. Note that this behavior is valid, if the bifurcation supports droplet
routing (i.e., the resistances of the successor channels have to be coordinated with
the droplet resistances) and both successor channels have the same channel section
(which is generally the case for bifurcations).
Furthermore, also other active as well as passive droplet sorting mechanisms
can be implemented in the discrete model. For example, if the payload and header
droplets consist of different volumes, the discrete model supports the sorting by the
droplet size in a discrete fashion.
101
and, afterwards, it is shown how an instance of the model for a given microfluidic
network can be derived (Sect. 8.1.2). An evaluation of the precision of the model is
provided in Sect. 8.1.3.
8.1.1 Definition of the Discrete Model
The proposed discrete model is based on the following main concepts: (1) a discrete
representation of time, (2) a distinction between payload and header droplets,
(3) a discrete consideration of droplet behavior at bifurcations and sorters, and
(4) constraints restricting the distance of droplets to avoid coalescences of droplets.
These concepts are briefly discussed next before the resulting model is illustrated
using an example.
Discrete Representation of Time: The model discretizes the continuous time during
the droplet flow into atomic time steps. This allows to describe the duration a droplet
requires to flow through a channel or to execute a module’s operation in terms of a
number of time steps.
Distinction between Payload/Header Droplets: Since the payload droplets and the
header droplets can be of different volumes and fluids, they can cause different resistances in the channels and modules. Accordingly, the discrete model differentiates
between these droplet types. More precisely, the number of time steps a payload
droplet requires to flow through a channel c ∈ C or to execute a module m ∈ M
is defined by the function pSteps : C ∪ M → N. Accordingly, the function
hSteps : C → N defines the respective number of time steps for header droplets
to flow through a channel c ∈ C. Hence, depending on the type, a droplet takes a
certain amount of time steps before it enters the succeeding channel or module in
the network.
Behavior at Bifurcations/Sorters: The physical behavior at bifurcations for passive
droplet routing is abstracted by taking the number of time steps into account which
is required for the respective droplet to flow through either of the successor channels.
More precisely, a droplet flows into the successor channel requiring the least amount
of time steps. If this channel already contains a droplet, it flows into the other
successor channel. This behavior is accordingly applied if both channels already
contain a droplet. Note that this behavior is valid, if the bifurcation supports droplet
routing (i.e., the resistances of the successor channels have to be coordinated with
the droplet resistances) and both successor channels have the same channel section
(which is generally the case for bifurcations).
Furthermore, also other active as well as passive droplet sorting mechanisms
can be implemented in the discrete model. For example, if the payload and header
droplets consist of different volumes, the discrete model supports the sorting by the
droplet size in a discrete fashion.