1
1
2
3
1
1 Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw,
Poland; 2 Division of Biological Science, Graduate School of Science, Nagoya University,
Nagoya, 464-8602, Japan, 3 Institute of Fundamental Technological Research, PAS, Pawińskieg o
5B, 02-106 Warsaw, Poland, * e -mail: garst@ichf.edu.pl
Below we derive the equality of the pressure drops
 p in the distribution channels. The diagram in Fig. S1 shows a simplified version of the network with just two outlet channels. Each of the three inlets (for phase A, phase B and the continuous liquid C) splits into two microchannels. Each michrochannel in the network is characterized by a set of parameters, such as its hydraulic resistance ( r ) and flow rate ( q ) marked with an apropriate index. We also denote the pressures ( p ) at confluent points, as marked on the diagram.

Fig. 12 Schematic diagram of the microfluidic system responsible for distribution of the inflow of two miscible streams A and B

The Kirchhoff’s circuit laws allow us to calculate the pressure ( p ) and flow ( q ) in each channel.
(p
A
–p
B1
)+(p
B1
–p
2
) – (p
C
–p
2
)+(p
C
–p
1
) – (p
A1
–p
1
) – (p
A
–p
A1
) = 0
(p
B
–p
A1
)+(p
A1
–p
1
) – (p
C
–p
1
)+(p
C
–p
2
) – (p
B1
–p
2
) – (p
B
–p
B1
) = 0
(p
C
–p
1
) + (p
1
–p
0
) – (p
2
–p
0
) – (p
C
–p
2
) = 0
From the Ohm’s law: r
Xi=
∆p/q
Xi where r
Xi
– the hydrodynamic resistance of the i -th channel, q
Xi
– the volumetric flux of X solution in the i -th channel,
∆p
– the pressure difference between two points in the same channel, we receive: r
A2 q
A2
+r p
(q
A2
+ q
B2
) – r q
C2
+r q
C1
– r p
(q
A1
+ q
B1
) – r
A1 q
A1
= 0 r
B1 q
B1
+r p
(q
A1
+ q
B1
) – r q
C1
+r q
C2
– r p
(q
A2
+ q
B2
) – r
B2 q
B2
= 0 r q
C1
+ r t
(q
A1
+ q
B1
+ q
C1
) – r t
(q
A2
+ q
B2
+ q
C2
) – r q
C2
= 0
After few algebraic transformations, we obtain: q
A1
= (qr + Q
A r
A2
– Q
B r
B2
)/( r
A1
+ r
A2
+ r
B1
+ r
B2
) q
B1
= (q(r
A1
+ r
A2
+ r
B1
+ r
B2
– r) – Q
A r
A2
+ Q
B r
B2
) / (r
A1
+ r
A2
+ r
B1
+ r
B2
) q
C1
= q
C1
= Q
C
/2
Where q = q
Ai
+ q
Bi
and 𝑄
𝑋
= ∑ 𝑛 𝑖=1 𝑞
𝑋𝑖
(information in main text). From calculation of
∆p
1
and
∆p
2
∆p
1
= p
1
– p
0
= r t
(q
A1
+ q
B1
+ q
C1
) = p
0
+ r t
(q + Q
C
/2)
∆p
2
= p
2
– p
0
= r t
(q
A2
+ q
B2
+ q
C2
) = p
0
+ r t
(q + Q
C
/2) we obtain
∆p
1
= ∆p
2
which indicates that
∆p is equal in all distribution channels. In general, for N channels for A and B streams we can introduce relations: r
A(i+1) q
A(i+1)
+ r p
(q
A(i+1)
+ q
B(i+1)
) – r q
C(i+1)
+ rq
Ci
– r p
(q
Ai
+ q
Bi
) – r
Ai q
Ai
= 0 r
Bi q
Bi
+ r p
(q
Ai
+ q
Bi
) – r q
Ci
+ r q
C(i+1)
– r p
(q
A(i+1)
+ q
B(i+1)
) – r
B(i+1) q
B (i+1)
= 0 r q
Ci
+ r t
(q
Ai
+ q
Bi
+ q
Ci
) – r t
(q
A(i+1)
+ q
B(i+1)
+ q
C(i+1)
) – r q
C(i+1)
= 0 where 𝑛 ∈ [1, 𝑛]
From these equations, we receive q
Ai
= (q r + Q
A r
A(i+1)
– Q
B r
B(i+1)
)/( r
Ai
+ r
A(i+1)
+ r
Bi
+ r
B(i+1)
) q
Bi
= q - q
Ai q
Ci
= q
C(i+1)
= Q
C
/n because 𝑄 𝑐 then
= ∑ 𝑛 𝑖=1 𝑞
∆p i
= p i
– p
0
= r t
(q
Ai
+ q
Bi
+ q
Ci
) = p
0
+ r t
(q + Q
C
/n)
∆p
(i+1)
= p
(i+1)
– p
0
= r t
(q
A(i+1)
+ q
B(i+1)
+ q
C(i+1)
) = p
0
+ r t
(q + Q
C
/n) yielding the expected equality
∆p i
= ∆p
(i+1)

Fig. 13 The layouts of the components of linear (upper set of graphics – (a), (b), (c)) and logarithmic (lower set – (d), (e),
(f)) gradient generators. Top layer (a, d) as well as bottom layer (c, f) comprise network of channels while inner layer (b, e) is equipped with holes to provide connection top and bottom layer with each other

Fig. 14 The dependence of R
2
from the rate of flow of the continuous phase in the case of the device generating gradient of content basing on fluids of different viscosities. Symbols on the plot code for different values of the rate of flow of the droplet phase Q
D
(squares 2, triangles 4, circles 8, and stars 16 [mL/h])