8.1 Discrete Model
103
and a header droplet) at various time steps. By this, the droplet flow through
a bifurcation is abstracted. Note that all involved droplets satisfy the distance
constraints, i.e. have a distance of at least T = 6 time steps (applies for both
hSteps and pSteps). All other components like channels not being part of a
bifurcation or modules can be represented and simulated in a similar fashion.
8.1.2 Determination of a Discrete Model Instance
The model proposed above allows to determine a discrete representation of
microfluidic networks using passive droplet routing. To this end, for a given
network, a designer needs to (1) define the “real world” time of an atomic time
step, (2) determine the functions pSteps and hSteps for each channel c ∈ C and
module m ∈ M, and (3) determine the minimal distance T of time steps between
droplets. By this, a model instance can be derived which allows to simulate the flow
of droplets in a discrete manner.
The first step is the responsibility of the designer who, by choosing the “real
world” time of an atomic time step T a (in [s]) defines the resolution (and, hence,
also the precision) of the model instance. With this information together with the
specification of all components, the pSteps-function can be determined. This is
done by performing the following steps for all channels c ∈ C and all modules
m ∈ M, i.e. for all entities e ∈ C ∪ M:
• Assume a single payload droplet flows in the considered entity e. Furthermore,
assume the microfluidic device does not contain any further droplets.
• Determine the droplet speed v d in the currently considered entity e by applying
the 1D analysis model and solving the equation system defined by the Kirchhoff’s
laws (cf. Sect. 3.2).
• Use the resulting speed together with the respective length of the entity to
determine the duration dur a payload droplet takes to flow through the channel
or to execute the module, i.e. determine dur =
l
v d
.
• Abstract the resulting duration dur to the corresponding discrete amount of time
steps, i.e. set pSteps(e) := [
dur
T a
].
The same is similarly conducted for a header droplet and its respective resistance in
order to determine the hSteps-function.
Example 8.2 Consider the bifurcation and its channel specification from Example 6.1 (cf. page 80). Following the steps from above (using an incoming flow rate
Q in = 0.6 μl/min, a header droplet length of 65 μm, a payload droplet length of
5 μm, as well as the fluid properties provided in Examples 6.1 and 6.2) allows to
determine the following droplet speed rates v d and, hence, durations dur for each
payload/header droplet and channel:
103
and a header droplet) at various time steps. By this, the droplet flow through
a bifurcation is abstracted. Note that all involved droplets satisfy the distance
constraints, i.e. have a distance of at least T = 6 time steps (applies for both
hSteps and pSteps). All other components like channels not being part of a
bifurcation or modules can be represented and simulated in a similar fashion.
8.1.2 Determination of a Discrete Model Instance
The model proposed above allows to determine a discrete representation of
microfluidic networks using passive droplet routing. To this end, for a given
network, a designer needs to (1) define the “real world” time of an atomic time
step, (2) determine the functions pSteps and hSteps for each channel c ∈ C and
module m ∈ M, and (3) determine the minimal distance T of time steps between
droplets. By this, a model instance can be derived which allows to simulate the flow
of droplets in a discrete manner.
The first step is the responsibility of the designer who, by choosing the “real
world” time of an atomic time step T a (in [s]) defines the resolution (and, hence,
also the precision) of the model instance. With this information together with the
specification of all components, the pSteps-function can be determined. This is
done by performing the following steps for all channels c ∈ C and all modules
m ∈ M, i.e. for all entities e ∈ C ∪ M:
• Assume a single payload droplet flows in the considered entity e. Furthermore,
assume the microfluidic device does not contain any further droplets.
• Determine the droplet speed v d in the currently considered entity e by applying
the 1D analysis model and solving the equation system defined by the Kirchhoff’s
laws (cf. Sect. 3.2).
• Use the resulting speed together with the respective length of the entity to
determine the duration dur a payload droplet takes to flow through the channel
or to execute the module, i.e. determine dur =
l
v d
.
• Abstract the resulting duration dur to the corresponding discrete amount of time
steps, i.e. set pSteps(e) := [
dur
T a
].
The same is similarly conducted for a header droplet and its respective resistance in
order to determine the hSteps-function.
Example 8.2 Consider the bifurcation and its channel specification from Example 6.1 (cf. page 80). Following the steps from above (using an incoming flow rate
Q in = 0.6 μl/min, a header droplet length of 65 μm, a payload droplet length of
5 μm, as well as the fluid properties provided in Examples 6.1 and 6.2) allows to
determine the following droplet speed rates v d and, hence, durations dur for each
payload/header droplet and channel: