Hardware Reference
In-Depth Information
the two droplets is no more than 7. The sufficiency of Constraint
1
and Constraint
2
is proven based on the analysis for Cases (2.a)
(2.c). When Constraint
1
and
Constraint
2
are satisfied, unwanted movement or splitting of droplets will not occur.
The number of all possible concurrent movements for the droplet pair is at least 10,
and each droplet can be moved along any feasible direction in the array.
Based on the above discussion, the necessity and sufficiency of Constraint
1
and
Constraint
2
in Lemma
6.1
are proven.
Note that Constraint
1
and Constraint
2
also ensure that when one droplet is
split, the other droplet does not undergo any unwanted splitting or movement. Since
the movement of a droplet from one electrode to its adjacent electrode is the basic
operation for mixing, dispensing, and transportation operations, Constraint
1
and
Constraint
2
can ensure the maximum degree of freedom for any concurrent fluid-
handing operations involving any two droplets.
6.3
ILP Model for Pin-Assignment
In this section, we develop an integer linear programming (ILP) model to optimally
solve the optimization problem for pin-assignment. As explained later, the model
forms the basis for evaluating heuristic solutions. On an M
N electrode array, let
x
i;m;n
be a binary variable defined as below.
8
<
1; if Pin i is connected to the electrode at the
m
th
row and n
th
x
i,m,n
D
column of the array
:
0; otherwise
where 1
i
L. The parameter L is the maximum possible index for the number
of control pins. The value of L can be set to an easily-determined loose upper bound
(e.g., M
N ).
The index of the control pin connected to the electrode at the m
th
row and n
th
column of the array is defined as P
m,n
. It can be expressed as P
m,n
D
P
i D1
i
x
i,m,n
.
The total number of pins that are assigned to electrodes is equal to the maximum
value of P
m,n
(where 1
m
M and 1
n
N ). Hence, the total number
of pins assigned to electrodes in the layout (i.e., N
tol
) can be written as: N
tol
D
Max
1i L;1mM;1nN
f
i
x
i,m,n
g
. Since the target of the ILP model is to derive
a feasible pin-assignment configuration that has the minimum number of control
pins. For a pin-limited digital microfluidic biochip, the objective function of the ILP
model is defined as:
minimize: N
tol
(6.1)
Next we map Constraint
1
and Constraint
2
of Sect.
6.1
into inequalities of the
ILP model. The electrode group whose central electrode is at the m
th
row and n
th
