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Predictable microfluidic mixing using discrete element microfluidics

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Predictable microfluidic mixing using discrete element microfluidics

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Content Predictable Micro
uidic Mixing using Discrete Element
Micro
uidics
Bryant Thompson
August 2015
The degree being conferred is: Master of Science
(DEPARTMENT OF BIOMEDICAL ENGINEERING)
FACULTY OF THE
UNIVERSITY OF SOUTHERN CALIFORNIA
1
1 Abstract
A novel micro
uidic platform has been developed which utilizes a library of standardized, modular
components manufactured using stereolithography. Large-scale 3-dimensional assemblies that
were previously dicult or impossible to construct in a planar orientation are now made realizable
by this system. Simple network analysis techniques allow for system predictability down to a
module-by-module basis. I herein propose a statistical approach to analyzing the performance
and predictability of micro
uidic mixing in the micro
uidic platform. Optimal mixing performance
is an essential component to almost all bioanalytical assays. Therefore, this study will allow for
the expansion of massively parallelized micro
uidic elements to conduct mixing for applications
in bio-analysis and general biochemical procedures.
2
Contents
1 Abstract 2
2 Background 4
2.1 Origins of Micro
uidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Characteristics of Flow in Micro
uidic Devices . . . . . . . . . . . . . . . . . . 4
2.2.1 Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Modularized Micro
uidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Discrete Elements for 3D Micro
uidics 7
3.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Hydraulic Analogy to Electronics . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Design Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Resistance Validation via Tunable Mixing . . . . . . . . . . . . . . . . . . . . . 12
3.5 Recongurable Droplet Generator . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6 In-Situ Monitoring of Droplet Generation . . . . . . . . . . . . . . . . . . . . . 21
4 Predictable Mixing using Discrete Element Micro
uidics 23
4.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Resistor-Based Component Library . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Rened Resistor Tolerance Approximation . . . . . . . . . . . . . . . . . . . . . 26
4.4 Parallel and Series Mixing Circuit Topologies . . . . . . . . . . . . . . . . . . . 28
4.5 Network Analysis of Circuit Topologies . . . . . . . . . . . . . . . . . . . . . . 29
4.5.1 Analysis of 2-input Fork Topology . . . . . . . . . . . . . . . . . . . . . 29
4.5.2 Analysis of 3-input Fork Topology . . . . . . . . . . . . . . . . . . . . . 30
4.5.3 Analysis of 3-input Ladder Topology . . . . . . . . . . . . . . . . . . . 31
4.6 Statistical Determination of Operational Tolerance . . . . . . . . . . . . . . . . 32
4.7 Mixing Law Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7.1 Deriving Volume Fraction from 2-input Fork Experimentation . . . . . . 38
4.7.2 Deriving Volume Fraction from 3-input Fork Experimentation . . . . . . 39
4.7.3 Deriving Volume Fraction from 3-input Ladder Experimentation . . . . . 39
5 Future Work 40
5.1 Surface Modication of Discrete Micro
uidic Components for Monodisperse Dou-
ble Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Active Components: Optical Sensor . . . . . . . . . . . . . . . . . . . . . . . 41
6 Acknowledgments 43
7 References 44
3
2 Background
2.1 Origins of Micro
uidics
In search of an optimal method for developing a gas analyzer able to instantly quantify dierent
gases and vapors, researchers at the Stanford Electronic Laboratories at Stanford University in
1979 turned to principles established in photolithographic methodologies to create a miniaturized
device built on foundations of gas chromatography (GC) [1]. Moving forward a decade, the
emergence of the term micro
uidic is rst brought to light under the Swedish Pharma company
Pharmacia Biosensor AB where again photolithography was used to fabricate a miniaturized sen-
sor to now quantify kinetics of monoclonal antibody-antigen reactions [2]. Like this, the science
of being able to manipulate low volumes of liquids (10
9
to 10
18
liters) within channels of ten
to hundred of microns in dimension [3] came to fruition. Having gained its roots from the inte-
grated circuit revolution, micro
uidic technology has advanced to a near similar state as that of
large scale integration seen in integrated circuits (IC) and has spread to several sectors including
molecular analysis, analytical chemistry, materials synthesis, and biomedical research [3{7].
Micro
uidic device fabrication has seen several cycles of ingenuity over the past few decades.
However, an approach remains to be introduced that allows for extremely facile construction of
micro
uidic assemblies consisting of discrete components that allow for rapid network analysis
and system characterization. While eective, the classic approach of utilizing photolithographic
methods for device fabrication is a time consuming, labor intensive, and costly process which
ultimately yields devices that are limited to two-dimensions [8]. Then micro
uidics grew ram-
pantly with the inception of rapid prototyping of micro
uidic systems in an elastomeric material,
poly(dimethylsiloxane) (PDMS) [9, 10]. In the work by Duy et al., authors demonstrated a
method of designing and fabricating a micro
uidic system out of PDMS in less than 24 hours.
Though PDMS has proven to resolve many common limitations in micro
uidic fabrication over
the past decade, it too has limitations in micro
uidic fabrication and device assembly. Ultimately,
the introduction of micro
uidics has motivated the advancement of three areas: novel approaches
for the fabrication of
uidic devices, development of components that can ease assembling com-
plex
uidic devices, and the physical behavior of
uids in micro
uidic systems [11]. Here, all three
areas of interest are explored.
2.2 Characteristics of Flow in Micro
uidic Devices
2.2.1 Navier-Stokes
Micro
uidic devices are very unusual as
ow at such small scales behave dierently than
ow
in larger scale systems [6]. Key physical properties that dene micro
uidic devices are brie
y
surveyed here, with which a foundation is established for the bulk of studies to follow.
For
ows of incompressible
uids, typically water and aqueous solutions, velocity elds can be
expressed by considering the incompressible Navier-Stokes equation for a uniformly viscous New-
tonian
uid, composed of terms that describe convective force, internal pressure force, and viscous
force, respectively:

@~ u
@t
+u +ru

=~ ur~ urp +r
2
~ u (1)
4
which can be reduced to a simplied Stokes equation in systems where inertial forces are negligible,
as in micro
uidic systems, allowing for the removal of nonlinear terms:

@~ u
@t
=~ ur~ urp +r
2
~ u (2)
Here, ~ u, m s
1
, is dened as ~ u = ~ u(~ r;t), the velocity eld of a
uid at a particular time, t,
in space, ~ r. Moreover, the
uid density, , is described in terms of kg m
3
, the viscosity, ,
is described in terms of Pa s, and p is the pressure in terms of Pa. If we now consider an
innitely long cylindrical channel with a
ow that has converged to steady state, the convective
and internal forces from Eq. 1 are reduced to zero, and Eq. 2 becomes:
rp =r
2
~ u (3)
which shows a balanced relationship between the net pressure of a system and the net viscous
force attributed to it. Flow within a pressure-driven system is termed Poiseuille
ow, which follows
a parabolic velocity prole across the cross section of a channel, such that u = 0 at r =R [11]
gives
ru =
R
2
r
2
4

dp
dx

=u
max

1
r
2
R
2

(4)
The Pousielle Flow, Eq. 4, is spatially integrated such that Hagen-Poiseuilles law is described as
Q =
R
4
8
p
L
(5)
where the volumetric
ow rate, Q (m
3
s
1
) , in a circular channel of length L (m), is described
in terms of the pressure loss, p, the radial cross section, R (m), and the dynamic viscosity
(Pa s).
2.2.2 Reynolds Number
Perhaps the most recognized dimensionless number in micro
uidics is the Reynolds number (Re),
an expression that characterizes the relationship between inertial forces to viscous forces in a

uidic system (Eq. 6).
Re =
f
intertial
f
viscous
=
U
0
L
0

(6)
Here, is the density of the
uid, U
0
the velocity, L
0
the travelled length of the
uid, and the
dynamic viscosity in terms of Pa s. In the case of micro
uidics, systems express low Re values,
which comes as a result of viscous forces dominating the inertial forces of
uid
ow, causing the
characteristic laminar
ow seen in micro
uidics. Low Re values leads to a simplication of the
Navier-Stokes equation by removing the nonlinear terms of the equation leading to the Stokes
equation (Eq. 2).
2.3 Modularized Micro
uidics
With more recent focus set on developing micro
uidic platforms that allow for rapid device as-
sembly, several pieces of work are herein highlighted to demonstrate previous approaches taken
5
towards creating modularized micro
uidic platforms. In the work by Shaikh et al., authors devel-
oped a modular micro
uidic platform whose architecture consisted of chip modules that can be
left as a single level or stacked onto one another to give multiple chip modules. At the center
of chip assemblies is a micro
uidic breadboard module that hosts the electromechanical control
structures (active components) of assembled systems. Fluidic channels in chips are interconnected
and can be customized for particular tasks [12]. As a variation of this, the work of Burns et al.
introduced modular 'blocks' that are PDMS slabs (made via soft lithography), which contain
particular channel routings, limited to two-dimensions, specic to the purpose the component
serves. Blocks are then laid down side by side, in a fashion similar to jigsaw pieces, such that
channels are aligned to create continuous
ow path [13]. Similar to the work of Shaikh et al.,
a breadboard system and modular micro
uidic components was developed via stereolithography
wherein
uidic components were plugged onto the breadboard base layer and connections between
components are made with custom H-shaped microchannel inserts [14]. Lastly, as a renement
to the work of Yuen et al., Jiang et al. introduced modular
uidic components made of PDMS
via soft lithography techniques that interconnect with plastic tubes. The renement here over
previous work is the characterization of components as resistors, done so by lumping together
parameters of individual PDMS components [15]. This analysis is not necessarily advantages as
components do not follow a standardized footprint and modeling isn't reduced to an element
level.
Here, a system of discrete modular micro
uidic components is fabricated via stereolithography,
such that connections between components are self registered and reversible, assemblies and
channel routing are not limited to two-dimensions. Moreover, fabricated components follow a
standardized geometric footprint allowing for lattice constructions, and in the latter portion of
this work terminal characteristics of components and assemblies are well dened by a library of
components dened by resistance that is validated by precision mixing of
uids.
6
3 Discrete Elements for 3D Micro
uidics
3.1 Background and Motivation
Micro
uidic technology has made great strides in
uid-handling sciences, but has done so mainly
in a fashion limited to 2-D congurations as a result of following in the footsteps of the semicon-
ductor industry. The standard planar orientation seen in integrated circuits may not necessarily
translate to hydraulic systems that require a sort of intricacy that comes unnatural to planar
geometries. This becomes immediately evident in complex hydraulic networks where routing in-
creases the design complexity for systems to be made realizable on planar silicone substrates. Not
only this, but increased system complexity in turn also increases the required spatial templates,
resulting in bloated costs and increased manufacturing diculty. Rather than following into the
paradigm that micro
uidics has evolved around for the last several decades, a platform has been
developed in which self aligning discrete components can be quickly connected in a reversible
manner to create fully customized micro
uidic systems free of clean-room processing. Analogous
to electronic elements that invoke standardized interconnect footprints, a library of components
was developed to give users the freedom of building robust systems that exhibit hydraulic char-
acteristics analogous to circuit theory that allow for network analysis. Now the task of modifying
hydrodynamic characteristics of micro
uidic elements has been reduced to simple modication
in computer-aided design (CAD) which are translated to fabrication via additive manufacturing.
Previously, modularized micro
uidics had been suggested in an eort to redirect the pattern of
fabrication system processing to a methodology that is more cost ecient and requires less fo-
cus on spatial design. In general, two strategies have been propose: interconnecting monolithic
chips [15{17] and systems in equivalence to electronic breadboards [12,14]. Other methods have
been introduced that make micro
uidics comparable to jigsaw pieces with self-aligning channels
that assemble in a chip-like fashion [13, 18]. Though in their best eorts, these systems oer
only the facade of benets promised from truly modularized discrete components analogous to
electronics as the core manufacturing methodology is predicated on fabrication by photolithogra-
phy. Here, an entirely new micro
uidic design is built on a platform of standardized components
fabricated using high-precision additive manufacturing techniques.
3.2 Hydraulic Analogy to Electronics
In its quick evolution, micro
uidics succeeded in moving forward in several branches of life sci-
ences but left behind the convention of fabricating a platform with the ability to perform network
analysis in a discrete manner due to highly complex and often times unstructured constructs.
While computational
uid dynamic (CFD) analysis provides methods of circumventing the is-
sue, it remains that CFD approaches require expensive proprietary software that demand certain
knowledge in theoretical
uid dynamics to move forward [19]. While several researchers have
implemented methods, particularly from circuit theory, to simplify the network analysis of their
systems [15, 20, 21], these techniques have not been particularly attractive as they are built on
platforms that are not built from standardized parts and build processing techniques remain
restrained to the limitations of clean room fabrication. Rather than approaching micro
uidic
modularity from the angle of previous methods, the use of additive manufacturing was explored
7
in order to manufacture high-precision discrete components unrestrained by common limitations
of clean room fabrication, which have previously been shown to be an eective approach to re-
alizing micro
uidic resolution [22{24].
Here, we go back by developing a platform of discrete micro
uidic components that follow the

uidic analogy to Ohms law from circuit theory, Hagen-Poisueille's law, that holds true for a

ow with low Reynolds Number by an incompressible
uid that is pressure-driven [25]. Previ-
ously introduced, the Hagen-Poiseuille law (Eq. 5) distinguishes the relationship between pressure,

ow-rate, and resistance of a system. Intuitively, the law can be reduced to:
p =QR
h
(7)
such that the expression is analogous to Ohms law
V =IR (8)
where pressure p is equivalent to voltage drop and
ow rate, Q, follows the behavior of current
in electronics and the hydraulic resistance, R
hyd
(Pa s
3
m
1
), is simply
R
hyd;cylinder
=
8L
R
4
(9)
for a straight cylindrical channel. In this case, the resistance is determined for a square channel,
not a cylindrical channel, such that the resistance is expressed as
R
hyd;square
= 28:4
L
h
4
(10)
where resistance is dominated byh, the square channel side length (m), shown in Figure 1. This
allows for an overall system development strategy that is more closely based on the methodology
behind circuit element production and electronic circuit design.
8
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
60
70
80
R (mm)
Hydraulic Resistance (GPa s m
3
)

Channel Length = 24.4mm
Channel Length = 44.8mm
200 400 600 800 1000 1200
0
50
100
150
h (um)
Hydraulic Resistance (GPa s m
3
)

Channel Length = 6mm
Channel Length = 8mm
A B
Figure 1: (A) Hydraulic resistance for cylindrical channel simulated at two dierent
lengths: 24.4mm and 44.8mm. For the case of cylindrical channels, hydraulic resistance
is dominated by the channel radius, simulated here from 0.2- to 1-mm radii, as a steep
decay is shown for increased channel radii. (B) Hydraulic resistance for square channel
simulated at two dierent path lengths, 6mm and 8mm, in this case the path length of
a straight pass component and a connector, respectively. In this case, the resistance
is dominated by the cross sectional side length of the channel. Simulation is shown
for cross sectional side lengths ranging from 200-um to 1200-um to demonstrate quick
drop around 300-um.
3.3 Design Concept
An initial library of micro
uidic components was designed to contain common
uid manipulating
elements such as junctions, mixers, splitters, and chip-to-world interfaces. Components were built
to follow a standardized 1x1x1 cm cubic geometry, which allows for easy assembly of simple to
complex systems having to pay no concern to the spatial distribution of components. Moreover,
components were designed with cues on their surfaces, serving as visual symbols to help orient
the system designer for rapid assembly. Similar symbolic marker methodology is used in discrete
electric components as in the case of resistors, inductors and diodes. Interface elements (i.e
inlet/outlet components) were designed to t standardized 1/16" OD tubing, building towards a
platform that does not require proprietary interconnect solutions.
Male-Male ended connectors were designed to create reversible seals between male connector
ends and female ports on components. Self aligning male pins insure channel continuity between
the 1-mm side length connector channel to the 500- or 750- um channel side length of the com-
ponents (Fig. 2). Connectors channel side length was made larger than component channel side
length to reduce the eects of hydraulic resistance while assuring low Reynolds numbers. Table
1 summarizes the library of constructed components alongside their hydrodynamic resistances,
calculated with the use Eq. 10, as all components were constructed with a square channel ge-
ometry such that net resistance was varied by modulating channel side length and/or channel
path length (Tbl. 2). All components were built using the stereolithography additive manufac-
turing services of neline, a protolabs company. The material chosen, Somos
R
WaterShed XC
9
11122, is a photopolymer material that generates ABS like components upon print. In particular,
the material features an optically clear and colorless end product that is ideal for micro
uidic
applications where visual inspection is necessary.
Connector
Component
Cue
Element
Spacer
a b
c
10mm
B A
C
Figure 2: (A) CAD assembly of a male-male connector con-
nected to a component with female-type port and a 750-um
straight-pass element. (B) Actual image of two components
connected together via a connector. Flat surfaces on com-
ponents allow for easy visual inspection to assure correct
seal has been made between connect pin and component
port. Cylindrical spacer on connector shown to magnify
contents passing through it. (C) CAD example of chip-
to-world interface accomplished by connecting standardized
1/16" PEEK tubing to interface tting.
10
Element h (m) Nomenclature R (MPa s m
3
) R (MPa s m
3
)
Connector 1000 R
C;1000
227.20 223:1 5:5%
Straight Pass
500 R
SP;500
2726.40 2720:41 3:7%
750 R
SP;750
538.55 525:69 6:2%
1000 R
SP;1000
170.40 169:67 3:1%
L-Joint
500 R
LJ;500
2726.40 2720:41 3:7%
750 R
LJ;750
538.55 525:69 6:2%
1000 R
LJ;1000
170.40 169:67 3:1%
T-Junction
500 R
(TJ);500
1363.20 1360:21 3:7%
750 R
(TJ);750
269.28 262:85 6:2%
1000 R
(TJ);1000
85.20 4:835 3:1%
Mixer
635 R
M;635
16227.00 17708:04 4:2%
750 R
M;750
6395.30 6218:5 7:2%
1000 R
M;1000
1846.00 1838:1 3:1%
X-Junction
500 R
(XJ);500
1363.20 1360:21 3:7%
750 R
(XJ);750
269.28 262:85 6:2%
1000 R
(XJ);1000
85.20 84:835 3:1%
Interface 750 R
I
448.79 438:08 6:2
XT-Junction 750 R
XT;750
269.28 262:85; 6:2%
XX-Junction 750 R
XX;750
269.28 262:85 6:2%
IR Sensor 642.5 R
IR;642:5
999.95 993:57 0:99%
Table 1: A compiled list of the constructed components with their designed
cross-sectional length, h, and resulting designed resistance, R
Des
. Expected
resistance, R
Exp
, was determined by resolving the square channel resistance
equation, Eq. 10, with measuredh values, and presented here with percentile
SD.
11
Table 2: List of components with their respective names,
CAD drawings, physical cues, and equivalent circuit dia-
grams.
3.4 Resistance Validation via Tunable Mixing
In order to validate the accuracy of the hydraulic resistance calculation a 2-input 1-output circuit
was constructed in order to compare the
ow rate through each branch that is dictated by the
respective branch resistance. The system, shown in Figure 3, is setup with two resistances in
parallel, R and R
s
, each of which is grounded by a solution of Milli-Q water with colored dye
at the inlets and to solution via R
o
at the outlet. One branch remains as a reference branch
with constant resistance, R, and parallel to it a branch, R
s
, that can be tuned by modulating a
single component, R
select
, as all structural components are identical on both branches. In order
12
to determine the
ow rate division attributed to changes by R
select
,
ow rates of each branch,
originally generalized by Eq. 7, were determined (Eq.11 -13) and a ratio of the two
ow rates
(Eq.14 -15) produced the mixing ratio, m
o
, described by Eq. 16.
R =R
I
+R
TJ;750
+ 3R
C;1000
+R
L;750
+R
SP;750
=R
struct
+R
ref
(11)
R
s
=R
I
+R
TJ;750
+ 3R
C;1000
+R
L;750
+R
select
=R
struct
+R
select
(12)
R
o
=R
TJ;750
+R
C;1000
+R
I;750
(13)
Q
y
=
0P
R
s
(14)
Q
b
=
0P
R
(15)
m
o
=
Q
y
Q
b
=
R
struct
+R
ref
R
struct
+R
select
(16)
Dierent components of R
select
were then used to modulate mixing ratio so that the fraction of
resident widths of each dye steam entering the junction could be optically measured and compared
to the theoretical fraction expected by Eq. 16. As suggested by [26], If both input streams share
equal dynamic viscosity, then the ratio of resident widths can accurately represent the
ow rate
ratio. The comparison between theoretical and experimental mixing ratio is given in Figure 4,
and is shown to have strong agreement, validating the premise of the using resistance components
to modulate mixing ratio.
The principle of modulating an output mixture with a comparator system (2-input, 1-output)
that can be tuned to selection, can be expanded in ways previously dicult or impossible by clean
room fabrication due to a restricting planar geometry. This concept is shown in Figure 5 as the
simple comparator sub circuit is expanded to a 2-, 3-, and 4-outlet system with the integration
of a TJ, X, or XT component near the particular sub-circuit. A constant pressure source driven
system then allows for the analysis of each sub-circuit outlet system where output is a resistance
ratio driven system, where equivalent circuit diagrams are shown in Figure 6.
13
Q
b
R
s
R Q
y

P
Q
o

R
o

a b
R
select

R
ref

A B
Figure 3: (A) CAD assembly for the 2-input, 1-output system in which
R
select
is varied and concentration changes respectively with R
ref
as a
constant reference resistance. (B) Circuit diagram equivalent to CAD
assembly where R is the total branch resistance of the reference branch
andR
s
is the total branch resistance of the tuning branch, with R
select
included in calculation along with interface, connector, and T-junction
components.
14
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
m
0

R
ref
/R
select

Experimental
Model
Figure 4: Validation of experimental mixing ratio model in com-
parison to resident width ratio for reference branch to the re-
sistance modulated branch, R
s
. Several R
select
were chosen
and error bars represent the standard deviation over 12 optical
measurements. Inset micrographs depict sample images used
to determine the width of collinear streams
owing into the
junction for particular R
select
components .
15
a
b
c
d
e
f
B D F
A C E
Figure 5: The standardized library footprint allows for the expansion of single-
outlet sub circuits to be parallelized in construction in order to create tunable
mixers with (A/B) two, (C/D) three, and (E/F) four outlets. As in the simpler
2-input 1-output scenario, each sub circuit in the expanded constructs can be
tuned to perform particular mixing ratios by selecting a resistance component to
modulate the branch resistance to the desired mixture.
16
C
B
A
Figure 6: Circuit equivalents for (A) 2-, (B) 3-, and (C) 4-
outlet tunable mixer system. Each sub-circuit produces an
output solution,m
o;n
, that is driven by a negative displace-
ment pressure source, which allows for parallel sub circuits
to be treated as equivalent parallel resistances. Moreover,
the symmetrical construct with respect to each output al-
lows the select and reference resistors to be the only fac-
tors, apart from structural resistance, that play a role in
output mixture.
3.5 Recongurable Droplet Generator
A platform with complete modularity supports the ability to quickly recongure systems without
having to re-fabricate a device particular to the desired function. Two quickly interchangeable
droplet generating systems are demonstrated to show this principle: a T-Junction and Flow-
Focus droplet generator. The T-Junction layout, shown in Figure 7, utilized a single syringe
pump to drive two dye-bearing aqueous streams, which were mixed and sub-sequentially sheared
by a carrier oil stream to create the micro-droplets. The carrier stream was held constant at 1
ml h
1
, while the aqueous phase
ow rate was varied to show how modulating the introduction
17
of the aqueous phase to the carrier phase aected the size distribution of droplets (Fig. 8A). The
T-Junction system is expanded to a four outlet system, as shown in Figure 9, to construct a
parallelized assembly of T-Junction sub circuits for high-throughput applications. Here, the input
carrier phase is split into four streams where each is then intersected by an input aqueous phase,
eectively generating micro droplets in each of the four output branches. A 3-dimensional layout
is further exemplied by creating a droplet generator capable of producing even smaller droplets
by focusing the carrier phase around an aqueous phase
owing perpendicular to it (Fig. 10).
The
ow-focus device simply replaces the terminal junction of the T-Junction schematic with an
X-junction component so that two carrier phase streams focus around and squeeze the aqueous
phase to create droplets. The carrier phase was kept at a constant
ow rate of 5 ml hr
1
to
prevent coalescing of droplets near the outlet. The droplet length is again determined for the

ow-focus system and is shown to generate droplets whose size depend on the input aqueous
phase
ow rate (Fig. 8B).
a b
c d
Carrier Phase
Aqueous
Phase
Aqueous Phase
A
C
B
D
Figure 7: (A) CAD assembly and (B/C) actual T-junction
droplet generation device. (C) Collinear dye-bearing water
streams passing through a 3-D helical mixer and outputting
a mixed solution that is then (D) used to create droplets
from a shearing carrier oil phase. In this assembly all con-
nectors have a cross sectional side length of 1-mm and all
other elements are 750- m.
18
0
100
200
300
400
500
600
700
800
900
0 500 1000 1500 2000 2500
Droplet Length (µm)
Aqueous Flow Rate (µL/hr)
Carrier Flow Rate = 5000 µL/hr
0
500
1000
1500
2000
2500
3000
3500
0 200 400 600 800 1000 1200
Droplet Length (µm)
Aqueous Flow Rate (µL/hr)
Carrier Flow Rate = 1000 µL/hr
a
b
B
A
Figure 8: Droplet length measurement distribution deter-
mined by taking micrographs at the center axis of the exit
tubing, done for (A) T-Junction and (B) Flow-Focus sys-
tems. Error bars denote SD of 12 length measurements.
We note the smaller achievable droplet sizes for the Flow-
Focus device in comparison to the droplet distribution of
the T-Junction system. For both systems the components
contain a 750-um channels and all connectors have a 1-
mm channel side length.
19
a b
c
d
Carrier Phase
Aqueous Phase
A B
C
D
Figure 9: (A/B) Actual and (C/D) CAD assembly of a
3-D T-Junction assembly to create droplets. Here, cross-
sectional side length for all connectors is 1-mm and all
other components are 750- m.
20
a
Carrier Phase
Aqueous Phase
Carrier Phase
b
c
A B
C
Figure 10: (A) CAD schematic and (B/C) actual
ow-
focus construct. The terminal end of the T-Junction
schematic is replaced with an X-junction component for
which two carrier phase streams focus around an aque-
ous phase
owing through it. The system uses 750-um
components and 1-mm connectors.
3.6 In-Situ Monitoring of Droplet Generation
One major benet of being able to construct 3-dimensional components bearing
uidic channels
is the ability of integrating active elements, such as sensors and actuators, into the fabricated
components. This is demonstrated with the incorporation of a near-infrared emitter-receiver
pair that is housed into a custom IR sensor component, shown in Figure 11A. Custom design
features on the side of the component allowed the diodes to sit rmly in place while a beam
path intersected a channel with 642.5um cross-sectional side length. The IR sensor component
was placed downstream of a T-Junction droplet generator to determine the frequency and size
distribution of droplets intersecting the beam path (Fig. 11B). A
uorocarbon oil (Halocarbon
4.2) was set as the carrier phase and intersected an aqueous phase stream to create water-in-oil
emulsions. Water droplets that cross the infrared beam path absorb the infrared light far more
than the carrier oil, allowing for collection of an analog signal that dierentiates between water
and oil based a predetermined threshold value (Fig. 11C). After communicating across a micro
controller and converting the analog signal to a digital signal, the droplet length is calculated by
utilizing the average
ow velocity in the channel and duration of droplet across the beam path,
which was then directly compared to measurements taken by optical micrographs (Fig. 11D).
21
Results are in good agreement with each other as they fall within deviation of one another,
leading to the conclusion that integration of active sensing and feedback components can now
be integrated to micro
uidic constructs with much facility.
Figure 11: (A) CAD assembly of a straight pass channel of 642.5um cross-sectional
side length that runs perpendicular to the beam of a near-infrared (NIR) diode
emitter and phototransistor receiver. (B) The IR sensor is placed down one block
length downstream from the droplet generating t-junction such that the water-in-
oil emulsions absorb the beam as the path is intersected, in turn (C) generating a
periodic signal corresponding to droplet frequency and size. (D) Duration of travel
across beam path and corresponding
ow rate of droplet train was used to calculate
droplet lengths, which were then compared to optical micrograph measurements.
Flow rates for the carrier and aqueous phases were held constant at 5 ml h
1
and
2mlh
1
, respectively. Droplet length average was determined to be 421.22um
27.54um and 416.90um 16.36um by the NIR sensor and optical micrographs,
respectively.
22
4 Predictable Mixing using Discrete Element Micro
uidics
4.1 Background and Motivation
Micro
uidic concentration gradient generation has become an increasingly investigated topic for
its ability to mimic biological events and further shed light on biological phenomena. Naturally
occurring gradients in nature can govern the development of metastatic cancer [27], migration
of immune cells [28,29], cellular development [30], and characterization of cell toxicity from drug
gradients [31]. In typical lab settings, mixing is accomplished by the use of syringes, pipettes,
burettes, and other expensive tools that require manual and procedural methodologies. These
systems are commonly notorious for producing a considerable range of error depending on volume
size or method of operation, which can have greatly adverse eects on the nal desired mixing
concentration. As a result, a need arises for a simple to use hand held tool with precise and
predictable mixing of low volumes that remain insensitive to operator variability.
Traditionally, in vitro methods of generating gradients included the use of biological hydrogels
[32{34] and micropipette-generated gradients [35], which were limited to results that lacked
precision, controllability, and predictability. More recently, modern micro
uidic techniques have
allowed for the development of devices capable of serial dilution [25, 36, 37], parallel mixing
[37], two-layer linear dilutions [38], 3-D combinatorial mixing [39], and logarithmic concentration
gradients [36]. Inherent to their design methodology, these systems lack the ability to be rapidly
assembled into systems that generate dilution gradients that meet user specications. Previously,
a micro
uidic platform of discrete elements capable of self-aligning and connecting in a reversible
manner was introduced [40]. Here the platform is rened from the previously introduced system
with a library of discrete components dened by inherent hydraulic resistance which enables
for a more compatible analogy to linear circuit analysis. A platform of this nature frees users
from focusing their eorts on predetermining spatial distribution required by standard 2-D silicon
substrates and redirects attention to the simple selection of components to construct micro
uidic
dilution systems unhindered by complexity. In turn, a systematic 3-step development strategy
is presented: (A) Selection of resistance based components, (B) selection of circuit topology
to achieve desired mixture, and (C) initial modeling of each micro
uidic circuit with expected
manufacturing error to simulate operating space of the terminal characteristics. In the rst step,
components with particular hydraulic resistance are selected to achieve the desired output mixture
from input streams, where systems are not aected by user-to-user error as commonly occurs
with typical mixing methods. Dierent styles of mixing can be achieved depending on the circuit
topology chosen. That is to say, mixing can occur in parallel with parallel input streams, or
in series with adjacent input streams. Lastly, a simple to follow statistical method is used to
determine the operating range of chosen circuit topology with selected resistance components
and their manufacture error. Two circuit schematics are constructed and the dilution across one
branch respective to the system is determined in order to validate the operation of the models
against physical simulation.
4.2 Resistor-Based Component Library
As previously mentioned in Discrete Elements for 3D Micro
uidics, hydraulic resistance of a
channel can be nely tuned by channel path length with cross-sectional side length held constant.
23
Here, a rened library of resistance based components is constructed such that components, or
blocks, are dened by their hydraulic resistance (Tbl. 3). A component library of this nature
allows users to select their components in a manner similar to selecting resistors for an electrical
assembly. The library adapts the analogy of 'wires' used in the electronics community into in the
form of wire components that are two orders of magnitude below the least resistive component.
As previously described by Equation 10, resistance of a square channel can be modulated by path
length L or the cross-sectional side length h, assuming a constant dynamic viscosity, , of 1
mPa s. A standard resistance unit of 1 GPa s m
3
is denoted as 1G and constitutes the basic
resistance unit of the resistance based library, dened by a 6 mm path length and 642.5 m
cross sectional side channel length. Resistors beyond 1G were fabricated by snaking or coiling
path lengths within the blocks to satisfy the square channel resistance equation. In order for the
only resistance to be experienced by systems to be from the dedicated resistance components,
the wire components were fabricated to have a 0.01G resistance by extending the cross sectional
side length to 2.0317 mm, which still retains a laminar
ow at reasonable
ow rates (Re roughly
0.1 at 200 ml/h) in a 6 mm path length. Furthermore, port components used for interfacing to
out-of-world systems (e.g. pumps or syringes) were fabricated with a cross sectional side channel
length of 1.1425mm, creating a resistance of 0.05G. Wire-like components are dened as having
parasitic resistance, similar to the parasitic resistance in electronics that is neglected due to
its considerably low resistance dened as being 2-3 orders of magnitude lower than resistance
components.
24
Class Name CAD Model Nomenclature Designed R (G) Expected R (G) Expected Error (%)
Port
(h = 1142.5 um) Port
R
P
0.05 - -
R
in
3 3.018 10.797 %
Wire
(h = 2031.7 um)
Connector W
C
0.01 - -
Straight Pass W
SP
0.01 - -
L-Joint W
LJ
0.01 - -
T-Junction W
TJ
0.01 - -
Resistor
(h = 642.5 um)
Straight Pass R
1
1 2.304 7.662%
Snaked R
2:5
2.5 0.920 3.305 %
Helix
R
5
5 4.582 2.592 %
R
10
10 9.139 2.244 %
R
25
25 22.824 1.852 %
Table 3: Library of constructed resistance-based components particular to their designed hydraulic resistance by modulating
track length and channel side length (h). Expected Resistance and Error, dened by 2, were determined through Monte
Carlo Simulation. The error ensures that 95% of designed components will fall within the expected resistance tolerance.
25
Library of constructed resistance-based components particular to their designed hydraulic
resistance, with units denoted as G, short for GPa s m
3
. Components classied as resistors
house a 642.5 um channel side length, allowing for resistance to be modulated by channel path
length. Expected Resistance and Error, dened by 2, were determined through Monte Carlo
Simulation. The error ensures that 95% of designed components will fall within the expected
resistance tolerance. Wire class components, with channel side length 2031.7um, are considered
to be a parasitic resistance, signifying its resistance value to be 2-3 orders of magnitude less than
the resistance class components.
4.3 Rened Resistor Tolerance Approximation
Steriolithographic manufacturing methods have an inherent limitation on print resolution, par-
ticular to the mechanics of optics that dene xy and z resolution. Therefore, a certain amount
of error is to be expected in the width, w, length, L, and height, h, of component channels
attributed to by the print direction being in the z direction. The resistance of channels is thus
better approximated as a rectangular channel, with error accounted for in the two print axis, xy
and z, as described for by Eq. 17, the hydraulic resistance of rectangular channels derived by
solving the earlier presented Navier-Stokes equation with a Fourier Series method [41].
R
hyd;rectangle
=
12L
h
3
w
"
1
1
X
n;odd
192h
n
5

5
w
tanh

nw
2h

#
1
(17)
Plane tolerances, for xy and z, were determined for by fabricating a large quantity of library
components and optically measuring their cross-sectional side lengths (Fig. 12). Tolerances were
then used in a Monte Carlo simulation to more accurately predict the standard deviation, , of
hydraulic resistance attributed to each element in the resistance-based library. More explicitly,
the resistance of each channel in both planes for every component was determined by drawing
parameters w, L, and h from a psuedorandom distribution characterized by the tolerances de-
termined for the xy and z orientations. Each channel segment within a component contributing
to the resistance was then summed to give the expected resistance (Fig. 13), along with the
expected error that is dened as 2 for 5000 parameter drawings (Table. 3). The deviations for
every resistance component, termed here as the manufacturing tolerance, assures that 95% of
the fabricated components fall within the listed tolerance.
26
B
A
Figure 12: Measured cross-sectional side length of the (A)
xy and (B) z directions for resistance-based compo-
nents. Optical micrograph measurements determined xy
and z to be 659:98 12:47m and 652:49 4:24m,
respectively, for over 70 measurements in each direction.
27
B A
Δxy
Δz
Δxy
Δz
Figure 13: (A) Port opening with centered channel for a 5G component with
642.5m cross-sectional side length. Due to the mechanics of stereolithography,
precision in the xy and z direction is expected to vary. (B) Determined cross-
sectional side length for the xy and z direction, also dened as the print plane
and print axis, respectively, used to create normal distributions for which a Monte
Carlo simulation could draw pseudorandom values from and input into resistance
equation. Total component resistance was then approximated by determining the
resistance of segments respective to xy (green lines) and z (red lines), by
drawing parameter values from their respective distribution and then adding all
calculated segment resistances together.
4.4 Parallel and Series Mixing Circuit Topologies
Two circuit topologies are introduced to study and validate the Hagen-Poisueille's law with
the fabricated library: the Fork topology (Fig. 14) and the Ladder topology (Fig. 15). In the
Fork topology, each input stream, which has a particular resistive component in its branch, is
rst introduced in the input end of each branch, and ultimately meets with all other streams
at the joining junction before being output. As the system is driven by a negative-pressure
source, every branch experiences an equivalent pressure drop that makes the concentration of
each input at the output very simple to determine. Moreover, a system of this nature is source-
invariant, as the particular volume fraction of each input, at the output, is determined solely by the
resistive component in each branch, and not the external source driving the system. This becomes
more evident in the Mixing Law derivation, particular to the topology being investigated, where
the resultant volume fraction of each input, for all topologies, is characterized by the resistive
components of the topology. In the Ladder topology, each input branch, starting from from the
branch furthest from the negative-pressure source, acts in series with the mixing resistor and then
in parallel with the adjacent branch, when then can be clumped together and act as a single
branch that again acts in series with a mixing resistor and in parallel with the adjacent branch.
Again, this system is source-invariant leading for calculation of each input volume fraction at he
output easy to compute and allows users the ease of driving the system manually. In this study,
28
a 2- and 3-input Fork topology is reviewed as well as a 3-input Ladder topology.
. . .
R
3
R
2
R
1
R
N-1
R
N

Figure 14: Generalized Fork topology where each branch
resistance experiences the same pressure drop across each
branch. Mixture between inlets thus occurs in parallel be-
tween branches.
R
M
R
M
R
M

R
1
R
2
R
N-2
R
N-1
R
N

. . .
Figure 15: Generalized Ladder topology where mixing
occurs in a serial manner, such that adjacent parallel
branches serially reduce to what is eectively a 2-inlet fork
topology, where R
N
constitutes one branch and all other
branches constitute the other branch (derivation shown in
text).
4.5 Network Analysis of Circuit Topologies
Analogous to practices performed in circuit analysis, the selected topologies were analyzed through
nodal analysis, such that an expression can be formed for the output concentration of each input
with respect to other input lines, denoted here as the volume fraction,

. We begin by analyzing
the 2-input Fork topology and generate a resistor based expression for quick volume fraction
calculation.
4.5.1 Analysis of 2-input Fork Topology
We begin deriving the volume fraction of each inlet substance (inlet 1 and 2) by expressing
the volume fraction as a ratio between the respective inlet
ow-rate, Q, and the total
ow-rate
(Eq. 18 - 19).
29

1
=
Q
1
Q
1
+Q
2
(18)

2
=
Q
2
Q
1
+Q
2
(19)
The
ow-rate across each branch is then expressed by the Hagen-Poiseuille equation (P =
QR), given as the relationship between the pressure dierence of solution, P
0
, to the junction at
which the two streams converge, P
x
, and the resistance of the branch (Eq. 20 - 21).
Q
1
=
P
0
P
x
R
1
=
P
x
R
1
(20)
Q
2
=
P
0
P
x
R
2
=
P
x
R
2
(21)
By inserting Eq. 18 - 19 into Eq. 20 - 21, the volume fraction of each inlet substance is
reduced to a set of equations dened by the selected resistance components, eectively the mixing
laws for the 2-input topology (Eq. 22 - 23).

1
=
R
2
R
1
+R
2
(22)

2
=
R
1
R
1
+R
2
(23)
4.5.2 Analysis of 3-input Fork Topology
As the case for the 2-input Fork topology, the 3-input Fork topology follows the same equiv-
alent pressure paradigm, which allows for a simple reduction to resistor-based volume fraction
expressions. Again, we express

as a ratio of branch
ow-rate to the total
ow-rate (Eq. 24 -
26).

1
=
Q
1
Q
1
+Q
2
+Q
3
(24)

2
=
Q
2
Q
1
+Q
2
+Q
3
(25)

3
=
Q
3
Q
1
+Q
2
+Q
3
(26)
The Hagen-Poiseuille Law is then applied to each branch in order to reduce the
ow-rates to
an expression of driving pressure and resistance:
Q
1
=
P
0
P
x
R
1
=
P
x
R
1
(27)
Q
2
=
P
0
P
x
R
2
=
P
x
R
2
(28)
30
Q
3
=
P
0
P
x
R
3
=
P
x
R
3
(29)
Lastly, the
ow-rate expressions (Eq. 27 - 29) are plugged into (Eq. 24 - 26) to get the
volume fraction equations for each inlet of the 3-input Fork topology. Again, we note that volume
fraction is source invariant:

1
=
R
1
R
1
+R
2
+R
3
(30)

2
=
R
2
R
1
+R
2
+R
3
(31)

3
=
R
3
R
1
+R
2
+R
3
(32)
4.5.3 Analysis of 3-input Ladder Topology
We begin examining the Ladder topology in the same manner as for the previous systems by
expressing

as the ratio between branch
ow-rate to the total
ow-rate of the system:

1
=
Q
1
Q
1
+Q
2
+Q
3
(33)

2
=
Q
2
Q
1
+Q
2
+Q
3
(34)

3
=
Q
3
Q
1
+Q
2
+Q
3
(35)
Again,

of each input is reduced to expressions of resistances by use of Hagen-Poiseuilles
Law:

1
=

P
2
R
1

P
2
R
1

P
2
R
2

P
3
R
3
=
1
R
1
1
R
1
+
1
R
2
+

R
3
(36)

2
=

P
2
R
2

P
2
R
1

P
2
R
2

P
3
R
3
=
1
R
2
1
R
1
+
1
R
2
+

R
3
(37)

3
=

P
3
R
3

P
2
R
1

P
2
R
2

P
3
R
3
=

R
3
1
R
1
+
1
R
2
+

R
3
(38)
Here, =
P
3
P
2
. A relative pressure is found through further analysis, where two auxiliary
equations are used to describe the conservation of
ow in the node most near the outlet:
P
2
P
3
R
M

P
3
R
3
=Q (39)
P
2
R
1

P
2
R
2

P
3
R
3
(40)
31
so that is reexpressed as:
= 1 +R
M

1
R
1
+
1
R
2

(41)
Collectively, this gives an expression for the volume fraction of each input solution:

1
=
R
2
R
3
R
2
R
3
+R
1
R
3
+R
1
R
2
+R
M
(R
1
+R
2
)
(42)

2
=
R
1
R
3
R
2
R
3
+R
1
R
3
+R
1
R
2
+R
M
(R
1
+R
2
)
(43)

3
=
R
1
R
2
+R
M
(R
1
+R
2
)
R
2
R
3
+R
1
R
3
+R
1
R
2
+R
M
(R
1
+R
2
)
(44)
In this study, the rst branch of every topology runs a stock 0.34 M NaCl solution which
dilutes particular to the topology and resistances chosen. All other branches run DI water. For
this reason the focus of the volume fractions is particular to the R
1
branch of every topology,
whose mixing laws is summarized by Table 4.
Circuit Topology Mixing Law
2-Inlet Fork Mixer

=
R
2
R
1
+R
2
3-Inlet Fork Mixer

=
R
2
R
3
R
1
+R
2
+R
3
3-Inlet Ladder Mixer

=
R
2
R
3
R
2
R
3
+R
1
R
3
+R
1
R
2
+R
M
(R
1
+R
2
)
Table 4: General mixing rules for the 2- and 3-inlet Fork
topology, as well as the 3-input Ladder topology. The
nal dilution ratio,

, is then easily calculated by input-
ing resistor component values, and deviations, from the
library into the mixing law for the particular topology.
4.6 Statistical Determination of Operational Tolerance
As previously shown, there exists a tolerance in channel dimensions that are expected to cause
variations in system operation, which could ideally be approximated using simple error analysis
techniques. However, for increasingly complex systems, error analysis computation by hand
becomes increasingly dicult as systems grow larger. Here, the Monte Carlo analysis method is
again employed (Fig. 16) to closely approximate the operating range for each topology, due to
variation in resistance from fabrication error propagation. The rst step in statistical determining
the operating range consisted of applying Monte Carlo to create a simulated resistor bin, each
of which has an a respective distribution inherent to its channel variation. Again, Monte Carlo
is applied by pseudo-randomly selecting resistors from the simulated resistor bins and inputing
32
them into the mixing laws dened per topology. The resultant output is a distribution of volume
fractions for each topology, allowing users to know the expected concentration range experienced
by input solutions at the output mixture.
Length
Probability
Δxy
Length
Probability
Δz
Resistance
Probability
R
1

Resistance
Probability
R
2.5

Resistance
Probability
R
25

.
.
.
Volume
Fraction
Probability
2-1 Fork
R
hyd,rectangle
Mixing Laws
Simulated
Resistance
Measured Channel
Tolerance
Expected Topology
Operating Range
Volume
Fraction
Probability
3-1 Fork
Volume
Fraction
Probability
3-1 Ladder
Figure 16: A general schematic for the Monte Carlo approach to determine the

range for input branch 1 of each topology, which carries the stock NaCl solution to be
diluted. Initially, a large batch of resistance components is fabricated for which the cross
sectional channel side length of component channels is measured optically to create a
distribution for the xy and z printing planes. Values from the tolerance distributions
(normally distributed) are then pseudo-randomly selected as input parameters for the
hydraulic resistance equation respective to rectangular channels. A simulated resistor
bin is then created, such that each resistor has a respective distribution with tolerance
set at the deviation from the mean simulated resistance value. Again, resistance values
are pseudo-randomly drawn and input as parameters to the mixing laws, respective
to the circuit topology being investigated, resulting in a distribution of

values that
eectively stand as the expected operating range of the particular system.
4.7 Mixing Law Validation
The three circuit topologies investigated were constructed, (Fig. 17 - 19), with dierent resistor
combinations, (Table 5 - 7), in order to validate the simulated operational tolerance set fourth by
Monte Carlo analysis. Each topology was constructed with the selected resistor components, such
that inlet PEEK tubing, 24.4mm in length, of branch R
1
was interfaced with an NaCl solution
33
(0.34M), and other inlet branches were interfaced with Milli-Q water. Systems were driven by
connecting a syringe on the output end, and withdrawing solutions slowly so as to insure a
suciently low Reynolds number. After channels were primed, syringe barrel was interchanged
with a clean barrel to collect roughly 0.5 - 1mL of mixture. Osmolality of dilute NaCl was
measured with an Osmomat 3000 osmometer, which was then used to determine the volume
fraction for each resistance combination, for all topologies (shown at the end of this section for
each topology). Once a system was constructed, data was collected by this manner in triplicate.
Figure 20 is arranged to show the simulated and experimental volume fraction data as deviations
from the designed volume fraction, better dened as the volume fraction calculated by inputting
perfect resistors (assuming no variation from designed CAD model) into the mixing laws, repeated
for every resistor combination attributed to each topology. The plots conrm the experimentally
tested volume fraction of NaCl in output solution to fall within the simulated operating range for
each topology, and each resistor combination.
R
1
R
2

R
M

R
M

R
P

R
P

R
in

R
2
R
1

R
P

W
LJ
W
TJ

R
in

B C A
Figure 17: (A) Circuit diagram of a 2-input Fork topology
whereR
1
andR
2
are selectively chosen for desired output
mixtures. (B) Equivalent hydraulic circuit where only R
1
and R
2
the selected resistance components, contribute to
the mixing ratio and wire components, denoted as parasitic
resistance, have negligible eect on resistance. Note that
the current source symbol in (A) is represented by a syringe
withdrawing solution through the outlet in (C).
34
R
3

R
M

R
2
R
1

R
M

R
P

R
in

R
1
R
3

W
LJ

R
in
R
in

R
P

R
P

W
TJ

W
TJ

R
2

R
P

B
A
C
Figure 18: (A) Circuit diagram of a 3-input Fork topology
where R
1
, R
2
, and R
3
are selectively chosen for desired
output mixtures. (B) Equivalent hydraulic circuit where
onlyR
1
,R
2
, andR
3
, the selected resistance components,
contribute to the mixing ratio and wire components, de-
noted as parasitic resistance, have negligible eect on re-
sistance. Note that the current source symbol in (A) is
represented by a syringe withdrawing solution through the
outlet in (C).
35
R
M
R
M

R
1
R
2
R
3

R
in

R
in

R
in

R
M

W
LJ
W
TJ

R
2
R
1
R
3

R
P

R
P
R
P

R
M

W
TJ

R
P

A
B
C
Figure 19: (A) Circuit diagram of a 3-input Ladder topol-
ogy whereR
1
,R
2
, andR
3
are selectively chosen for desired
output mixtures. (B) Equivalent hydraulic circuit where
onlyR
1
,R
2
,R
3
, the selected resistance components, con-
tribute to the mixing ratio and wire components, denoted
as parasitic resistance, have negligible eect on resistance.
Note that the current source symbol in (A) is represented
by a syringe withdrawing solution through the outlet in
(C).
36
2-Inlet Fork Mixer
R
1
(G) R
2
(G) R
3
(G) Designed

Expected

Expected Error (%)
R
1
R
1
- 0.500 0.499 11.859 %
R
1
R
2:5
- 0.421 0.426 11.998 %
R
1
R
5
- 0.333 0.341 12.511 %
R
1
R
10
- 0.235 0.245 13.327 %
R
1
R
25
- 0.125 0.132 14.400 %
R
2:5
R
1
- 0.579 0.575 8.711 %
R
5
R
1
- 0.667 0.659 6.431 %
R
10
R
1
- 0.765 0.756 4.318 %
R
25
R
1
- 0.875 0.868 2.247 %
Table 5: Resistor combination for the 2-inlet Fork Topology, where the R
1
branch runs
a 0.34 M solution, and remaining branches run Milli-Q water, which are then withdrawn
into a syringe after being mixed in the system. The designed mixing ratio using as-
designed resistance values, Designed

, and expected mixing ratio from Monte Carlo
simulation that takes into account build error, Expected

, are calculated by using the
respective mixing law shown in Table 4. The Expected Error presented is two standard
deviation, 2, from the expected mixing ratio, the average value, initially determined
by manufacturing tolerance.
3-Inlet Fork Mixer
R
1
(G) R
2
(G) R
3
(G) Designed

Expected

Expected Error (%)
R
1
R
1
R
1
0.333 0.333 13.587 %
R
1
R
5
R
1
0.200 0.206 11.569 %
R
5
R
1
R
1
0.400 0.397 12.169 %
R
1
R
10
R
1
0.133 0.139 11.418%
R
10
R
1
R
1
0.433 0.430 11.930 %
R
1
R
2:5
R
1
0.366 0.365 12.636 %
R
2:5
R
1
R
1
0.266 0.270 12.754 %
Table 6: Resistor combination for the 3-inlet Fork Topology, where the R
1
branch runs
a 0.34 M solution, and remaining branches run Milli-Q water, which are then withdrawn
into a syringe after being mixed in the system. The designed mixing ratio using as-
designed resistance values, Designed

, and expected mixing ratio from Monte Carlo
simulation that takes into account build error, Expected

, are calculated by using the
respective mixing law shown in Table 4. The Expected Error presented is two standard
deviation, 2, from the expected mixing ratio, the average value, initially determined
by manufacturing tolerance.
37
3-Inlet Ladder Mixer
R
1
(G) R
2
(G) R
3
(G) Designed

Expected

Expected Error (%)
R
1
R
1
R
1
0.286 0.289 14.241 %
R
1
R
1
R
5
0.174 0.181 12.618 %
R
1
R
10
R
1
0.380 0.380 12.253 %
R
5
R
1
R
1
0.364 0.362 12.371 %
Table 7: Resistor combination for the 3-inlet Ladder Topology, where the R
1
branch
runs a 0.34 M solution, and remaining branches run Milli-Q water, which are then
withdrawn into a syringe after being mixed in the system. The designed mixing ratio
using as-designed resistance values, Designed

, and expected mixing ratio from Monte
Carlo simulation that takes into account build error, Expected

, are calculated by
using the respective mixing law shown in Table 4. The Expected Error presented is
two standard deviation, 2, from the expected mixing ratio, the average value, initially
determined by manufacturing tolerance.
Measured
Expected Upper Bound
Expected Lower Bound
Avg. Expected
c b a
Figure 20: Comparison of experimental mixing ratio deviation from designed mixing ratio in com-
parison to simulated mixing ratio deviation to designed mixing ratio. For each graph, the upper
and lower bound describe a 2 deviation from the expected mixing ratio, such that the shaded
region, eectively the simulated operating space, is established by the manufacture tolerance that
suggests 95 % of the fabricated resistor elements fall within specication. Experimental data lie
within the simulated operational working space for the (A) 2-inlet Fork Topology, (B) 3-inlet Fork
topology, and (C) 3-inlet Ladder Topology.
4.7.1 Deriving Volume Fraction from 2-input Fork Experimentation
We consider the output concentration and its corresponding
ow-rate to be equivalent to the

ow-rate and the concentration it carries across each branch, such that for the 2-input Fork
topology the following expression is formulated:
c
out
Q
3
=c
1
Q
1
+c
2
Q
2
(45)
which can be further reduced to:
38
c
out
=c
1

1
+c
2

2
(46)
Here, we assume the concentration c
2
to be equivalent to zero as, in all topologies, only the R
1
branch carries concentrate solution and all others run Milli-Q water. With this, an expression for
the volume fraction of branch R
1
can be written as:

1
=
c
out
c
1
(47)
where c
1
is the stock NaCl concentration.
4.7.2 Deriving Volume Fraction from 3-input Fork Experimentation
As in the case of the 2-Input Fork topology, the output concentration is again expressed as a
weighted sum of the concentration and the
ow rate that drives it:
c
out
Q
out
=c
1
Q
1
+c
2
Q
2
+c
3
Q
3
(48)
c
out
=c
1

1
+c
2

2
+c
3

3
(49)
In this case, c
2
and c
3
are equivalent to zero, so that

1
=
c
out
c
1
(50)
4.7.3 Deriving Volume Fraction from 3-input Ladder Experimentation
Lastly, the 3-input Ladder topology again reduces the volume fraction,

to a ratio of the output
concentration to the stock concentration:
c
out
Q
out
=c
1
Q
1
+c
2
Q
2
+c
3
Q
3
(51)
c
out
=c
1

1
+c
2

2
+c
3

3
(52)
where c
2
and c
3
are equivalent to zero, so that

1
=
c
out
c
1
(53)
39
5 Future Work
5.1 Surface Modication of Discrete Micro
uidic Components for Monodis-
perse Double Emulsions
As previously shown, there are many benets to a truly modularized micro
uidic platform in
comparison to classic fabrication of single layer PDMS slabs, which experience limitations in
build geometry and rapid device assembly. Moreover, the polymer structure of PDMS makes
it permeable to organic solvents [42] and prone to absorbing low-molecular weight molecules
in channel
uids [43]. This has led an abundance of research to be focused on methods for
long-lasting surface modication of pre-assembled micro
uidic devices. Previous methods have
included UV treatment [44], UV exposure to create discontinuous surface patterns via chemical
vapor deposition (CVD) that are limited to modifying single un-assembled devices [45], and sol-
gel surface coatings that alter channel geometry [46]. However, these techniques often make
use of harmful chemicals and methodologies that are limited in throughput [47]. More recently,
the work of Riche et al. demonstrated the use of initiated chemical vapor deposition (iCVD)
to pattern
uoropolymer coatings within assembled micro
uidic devices [47, 48]. A modularized
micro
uidic platform of discrete components is a suitable candidate for components modied at
the channel surface level to create complex systems containing components with dierence surface
coatings. We have already successfully shown that single components can be coated to increase
the hydrophobicity of channel surfaces (Fig. 21). In Figure 21, iCVD was used to coat channel
surfaces of a straight pass component with poly(1H,1H,2H,2H-per
uorodecyl acrylate-co-ethylene
glycol diacrylate), or poly(PFDA-co-EGDA), which eectively created a low surface energy lm
that increased hydrophobicity. The contact angle of a water droplet surrounded by oil in the
channel increased from 67.9

to 138.3

. Alternatively, 2-hydroxyethyl methacrylate (HEMA)
monomer could be cross linked with EGDA to deposit a thin lm layer of poly(HEMA-co-EGDA)
that would increase hydrophilicity of channel surfaces. Modularized micro
uidic components allow
us to coat batches of components at a time, highly increasing the throughput of surface modied
component channels. One common application that requires both types of surface characteristics
is double emulsion devices, which nd uses in food applications [49,50], personal cosmetics [51,52]
and in the pharmaceutical industry [53]. Double emulsion devices commonly incorporate highly
customized micro-capillary glass devices [54{57] that are dicult to fabricate, reproduce, and
require hand alignment that ultimately make it unsuitable for massive parallelization. PDMS
devices are also utilized, but again require fabrication via soft lithography and complex surface
modication techniques that slow down the manufacturing process at the device level [58, 59].
Here, a system of interchangeable discrete components with variable surface modications for
double emulsions generation is proposed (Fig. 22). Figure 22A shows a general schematic for
the function of a double t-junction micro
uidic device to generate double emulsions. Figure 22B
embodies the general schematic with modular components such that water-oil emulsions pass
through channels with low surface energy and then transfer to water-oil-water emulsions that
pass through channels with high surface energy.
40
112.1°
41.7°
b a
Figure 21: Contact angle is here measured between a wa-
ter droplet in two components: Water droplet surrounded
by an oil carrier stream in an (A) uncoated and (B) coated
component. Surface modication eectively shows the in-
creased hydrophobicity of channel walls via initiated chem-
ical vapor deposition.
Q
W,OL
Q
W,IL

Q
O

Inner Phase
Outer Phase
Middle
Phase
Hydrophobic
Coating
Hydrophilic
Coating
B
A
Figure 22: (A) General diagram of a double emulsion device such thatQ
O
is a carrier oil
stream that comprises the middle layer of a double emulsion,Q
W;IL
comprises the inner
aqueous core of a double emulsion andQ
W;OL
is an outer aqueous carrier stream. (B) An
equivalent CAD diagram of the proposed double emulsion device where droplets in the
carrier oil stream
ow through surface modied channels with increased hydrophobicity
(orange components), which then enter hydrophilic channels as the outer carrier phase
shears the incoming droplet containing oil stream (purple components).
5.2 Active Components: Optical Sensor
The facile integration of o-the-shelf electronics to discrete micro
uidic components is a hall-
mark of the previously introduced technology. As an expansion to previous work, the integration
41
of an optical sensor for UV/vis absorbance detection can be demonstrated to overcome di-
culties present in sensor integration for photolithographic manufactured devices. The range of
absorbance measurement applications is abundant, yet has seen limited integration into micro
u-
idic devices due to sensitivity issues at the micro channel scale and limited integration methods
via classic photolithography device fabrication [60]. As suggested in Figure 23 a component may
be fabricated to have a snaking 3-dimensional channel consisting of a material that will not re
ect
light, such that a light-emitting diode (LED) may integrated to one end of a straight channel
and emitted light is received on the opposing end by a photodiode. The channel carrying emitted
light to the receiver end would virtually eliminate sensitivity issues that are present in devices
where beam path is limited to a cross section of a microchannel.
A B
Figure 23: (A) Iso view of optical sensor component where custom designs house an
LED and receiver with viewing windows to allow for absorbency measurements. (B)
Close up look at the snaked channel element inside the component where a straight
channel of known length creates a center line path from LED light to the receiver on
the opposing end.
42
6 Acknowledgments
I would like to acknowledge:
Dr. Noah Malmstadt for accepting me into his group when I rst arrived at USC.
PhD Candidate Krisna Bhargava for working closely and teaching me during the past two years.
All the folks at my lab that continually provided advice and suggestions and made becoming a
member of the lab an easy task.
43
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47

Abstract (if available)

Abstract A novel microfluidic platform has been developed which utilizes a library of standardized, modular components manufactured using stereolithography. Large‐scale 3‐dimensional assemblies that were previously difficult or impossible to construct in a planar orientation are now made realizable by this system. Simple network analysis techniques allow for system predictability down to a module‐by‐module basis. I herein propose a statistical approach to analyzing the performance and predictability of microfluidic mixing in the microfluidic platform. Optimal mixing performance is an essential component to almost all bioanalytical assays. Therefore, this study will allow for the expansion of massively parallelized microfluidic elements to conduct mixing for applications in bio‐analysis and general biochemical procedures.

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Asset Metadata

Creator Thompson, Bryant (author)

Core Title Predictable microfluidic mixing using discrete element microfluidics

School Viterbi School of Engineering

Degree Master of Science

Degree Program Biomedical Engineering

Publication Date 07/21/2015

Defense Date 07/21/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 3D‐printed microfluidics,microfluidics,modular microfluidics,OAI-PMH Harvest,SLA

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Malmstadt, Noah (committee chair), McCain, Megan (committee member), Roberts, Richard W. (committee member)

Creator Email bryantth@usc.edu,bryantthompson05@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-601219

Unique identifier UC11301312

Identifier etd-ThompsonBr-3655.pdf (filename),usctheses-c3-601219 (legacy record id)

Legacy Identifier etd-ThompsonBr-3655-0.pdf

Dmrecord 601219

Document Type Thesis

Format application/pdf (imt)

Rights Thompson, Bryant

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA