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movement or splitting of droplets will not occur. The number of possible
concurrent movements in this case is 1
C
6
D
7, as explained below. Droplet
1 and Droplet 2 can be moved to the electrodes controlled by Pin B concurrently
(1 possible concurrent movement); Droplet 1 can be moved to the electrodes
that are controlled by Pins E, C, and D; while Droplet 2 can be moved to the
electrodes controlled by Pins G and H (3
2 possible concurrent movements).
(3.e) Pin B is connected to E
1C
or E
2C
, and Pin A is neither connected to E
1C
nor
E
2C
: This case can be analyzed in the same way as Case (3.d). The number of
all possible concurrent movements for the droplet pair is 7.
Based on the above analysis, we conclude that when two CEGs share two pins,
for Case (3.a) and Case (3.b), droplets may undergo unwanted movement. For Cases
(3.c)
(3.e), unwanted movement or splitting of droplet will not occur. However,
the number for possible concurrent movements is no more than 7. Comparing
with a direct-addressing biochip, the flexibilities of droplet movements in Cases
(3.c)
(3.e) are relatively low. Hence such pin assignments are not considered in the
biochip designs studied in this chapter.
Based on the above discussion, we obtain the following lemma, which provides
a necessary and sufficient condition for the acceptance of any arbitrary pin-
assignment configuration with two droplets on two arbitrary positions of the biochip.
The goal is to guarantee significant flexibility for the concurrent movement of
droplets, and also prevent unwanted splitting or movement of droplets.
Lemma 6.1.
Consider an electrode array with any arbitrary pin-assignment con-
figuration. Suppose two droplets are located at any two arbitrarily chosen elec-
trodes. The following constraints are necessary and sufficient to avoid any unwanted
splitting or movement of droplets, to permit the movement of each droplet along
any feasible direction in the array, and to guarantee that the number of possible
concurrent movements is no less than 10:
Constraint 1. In the same control electrode group, any two electrodes cannot be
connected to the same control pin.
Constraint 2. Any two non-overlapping electrode groups cannot share more than
one pin.
Proof.
The necessity of Constraint
1
can be proven by
reductio ad absurdum
.We
assume that two electrodes in the same CEG share one pin. Suppose unwanted
splitting and movement of droplets can be avoided, and each droplet can be moved
along any feasible direction in the array. From the discussion of Cases (1.a)
(1.c),
we can find that droplets may undergo unwanted splitting or movement, or the
movement of a droplet along some directions can never be achieved. Hence we
have reached a contradiction.
t
The necessity of Constraint
2
also can be proven by
reductio ad absurdum
.We
assume that two CEGs share k (k>1) control pins. From the discussion of Cases
(3.a)
(3.e), we find that in some cases, droplets may undergo unwanted splitting or
movement. In every other case, the number of possible concurrent movements for
