Thermal modeling and design analysis of a continuous flow microfluidic chip

Thermal modeling and design analysis of a continuous flowmicrofluidic chip

Sumeet Kumar, Marco A. Cartas-Ayala, Todd Thorsen*

Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

a r t i c l e i n f o

Article history:

Received 25 March 2012

Received in revised form

2 December 2012

Accepted 4 December 2012

Available online 12 January 2013

Keywords:

Thermal modeling

Microfluidic design

Continuous flow microfluidics

Thermocycling

Polymerase chain reaction (PCR)

a b s t r a c t

Although microfluidics has demonstrated the ability to scale down and automate many laboratory

protocols, a fundamental understanding of the underlying device physics is ultimately critical to design

robust devices that can be transitioned from the benchtop to commercial products. For example, the

miniaturization of many laboratory protocols such as cell culture and thermocycling requires precise

thermal management. As device complexity scales up to include integrated electrical components,

including heating elements, thermal chip modeling becomes an increasingly important part of the design

process. In this paper, a computationally efficient, three-dimensional thermal fluidic modeling approach

is presented to study the heat transport characteristics of a continuous flow microfluidic thermocycler

for polymerase chain reaction (PCR). A two-step simulation model is developed, consisting of a solid

domain modeling of the entire microfluidic chip that examines thermal crosstalk due to lateral diffusion

across multiple thermal cycles, and a one pass simulation model to study the thermal profile in the

fluidic domain as a function of critical parameters like flow rate and microchannel material. The results of

the solid domain model are compared against experimental measurements of the thermal profile in

a PDMS-glass microfluidic thermocycler device using a combination of thermocouples and an infrared

(IR) camera. The suitability of the device in meeting the ideal thermocycling profile at low flow rates is

established and it is further shown that higher flow rates lead to deterioration in thermocycling

performance. Thermofluidic modeling tools have the potential to streamline the physical microfluidic

device design process, reducing the time required to fabricate functional prototypes while maximizing

reliability and robustness.

� 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

Microfluidic systems have gained tremendous attention over

the past two decades with regard to their potential to automate

chemical and biological assays at a fraction of the cost and time of

traditional benchtop research. Microfluidic chips enable the mini-

aturization of assays and offer the possibility of performing

numerous experiments rapidly and in parallel, thus enhancing

throughput and reducing the overall cost and consumption of

reagents. Microfluidics has made important contributions to many

biological and medical fields, including enzymatic analysis [1], DNA

analysis [2], proteomics [3,4], nano-particle fabrication [5,6] and

drug delivery [7].

Thermal control is a critical element of many biological and

chemical assay systems, affecting processes like enzyme catalysis,

hybridization between biomolecules (nucleic acids, proteins), and

cell culture. Polymerase chain reaction (PCR) is one of the most

commonly used biochemical reactions that requires precise ther-

mocycling, making it a good choice for a model system to study

spatiotemporal heat transfer in miniaturized diagnostic platforms

such as microfluidic chips. Microfluidics and micro electro-

mechanical systems (MEMS) offer several advantages for PCR

over conventional thermocyclers, including faster thermal ramping

rates [8e10], reduced sample volumes [9,11], disposability [12e14],

portability [12,15], functional integration of sample preparation,

and post-PCR product detection [8,16].

The history of microfluidic PCR devices dates back to 1993 when

Northrup et al. [17] demonstrated the first silicon-based stationary

chamber PCR device. Since then, continued efforts have been

applied toward developing cheap, portable, reliable and on-field

applicable microfluidic systems for PCR. In general, microfluidic

thermocycling can be performed in two different ways: 1)

stationary; heating and cooling reactants in the same chamber and

2) continuous flow; heating and cooling reactants as they move* Corresponding author. Tel.: þ1 781 981 5227; fax: þ1 781 981 6179.

E-mail address: [email protected] (T. Thorsen).

Contents lists available at SciVerse ScienceDirect

International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

1290-0729/$ e see front matter � 2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.ijthermalsci.2012.12.003

International Journal of Thermal Sciences 67 (2013) 72e86

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through channels having an imposed temperature distribution.

Both types of architectures for microfluidic PCR have been previ-

ously developed and characterized by many research groups [18e

20]. In the stationary chamber design, a micro or nanoliter

chamber containing the PCR solution is cycled between different

temperatures. In contrast, continuous flow-type PCR chips follow

the ‘time-space’ conversion principle and typically consist of three

independent, fixed temperature zones in space with the PCR

sample continually flowing between them via a microchannel.

There are several advantages of the continuous flow architecture

over thermocycling within stationary chambers. Notably, temper-

ature transition times are minimized as the thermal inertia of the

system is minimized (with the only significant contribution due to

the thermal mass of the sample), and, adjusting the flow rate of the

samples, reaction volume can be scaled up from nanoliter to

microliter scale volumes, making the system suitable for down-

stream diagnostic applications.

As chip device size decreases, thermal crosstalk becomes an

important issue due to the temperature sensitivity of the reaction. A

central challenge in using microfluidic systems for thermocycling

applications is to quantify its thermal performance. Non-specific

temperature profiles in the microfluidic chip can lead to inefficient

reactions, and in some extreme cases, failed reactions. A compre-

hensive understanding of the heat transport mechanisms in the

microfluidic device is critical for making functional parts [21e27].

Unlike momentum and species transport analysis, which are

confined to the fluidic domain, thermal modeling in microfluidics

presents some unique challenges [28e32]. The presence of thermal

diffusion necessarily extends the modeling domain from the region

of interest (i.e. the fluid domain) to encompass the material

bounding the microchannels. In contrast to a macroscale system,

where the fluid domain is often of comparable size to the solid

regions, a microchannel system typically encompasses only a very

small fraction of the substrate and thus heat transfer is significantly

influenced by thermal diffusion process through the solid regions

that may lead to thermal crosstalk. Taking the millimeter-scale

physical dimensions of microfluidic chips into consideration, with

temperature gradients generated by proximal or embedded heating

elements, a conjugate, three-dimensional model becomes neces-

sary to completely capture lateral thermal diffusion, which strongly

affects the thermal profile in the fluid domain.

Three-dimensional conjugate heat transfer in microchannel

flows has been well studied especially in the context of heat sinks.

Earlier studies focused primarily on numerical implementation of

the three-dimensional conjugate heat transfer equations, typically

in a rectangular microchannel geometry extracted from a multi-

channel heat sink [33e36]. Recently, Nunes et al. [37] extended the

understanding of heat sinks by developing a 2D model of parallel-

plate microchannel geometry and comparing it with experiments.

They showed that conjugate heat transfer and fluid axial diffusion

leads to non-uniform local Nusselt number. Kosar [38] studied the

effect of substrate thickness in straight microchannel heat sinks by

implementing a 3D simulation and developed an empirical Nusselt

number correlation. Three-dimensional transient conjugate heat

transfer simulations have also been performed to study time-

dependent heating of rectangular straight microchannels [39].

In heat sinks, the principal objective is to remove heat from

a substrate using convective and conductive heat transport. Such

modeling has primarily addressed understanding and optimization

of the bulk cooling characteristics of heat sinks and the tempera-

ture distribution in the solid domain. For biochemical applications,

it is imperative to study the temperature profile in the fluid domain

as a function of design and operating parameters. Furthermore, for

many continuous flow designs, the serpentine configuration of the

microchannels makes it critical to capture the effect of thermal

crosstalk. Wang et al. [25] previously presented a two-dimensional

thermal fluidic model to predict the performance of a continuous

flowmicrofluidic chip. Though two device performance parameters

were defined to describe the uniformity of temperature and devi-

ation from target temperatures, limited studies were carried out to

understand variation in temperature profile with respect to design

variation and operating parameters. Similarly, Li et al. [40] devel-

oped a two-dimensional semi-analytical thermal transport model

and carried numerical simulation to predict temperature profile in

the continuous flow PCR microchip. Chen et al. [41] considered

a three-dimensional model of the chip to first estimate temperature

distribution in the solid domain but evaluated the temperature

distribution in the fluid domain using a simplified two-dimensional

model. Though some work has been done on understanding

thermal profiles in continuous flow architectures [21,25,40e42],

modeling efforts have been limited to simplified two-dimensional

geometry which neglects thermal crosstalk due to lateral diffu-

sion and the effect of convective heat transfer on device perfor-

mance has not been comprehensively studied.

In this paper, a detailed, three-dimensional thermal modeling of

a continuous flow microfluidic thermocycler is performed. Design

and fabrication of the microfluidic platform is first presented and

the implications of the channel geometry on residence time and

hydraulic resistance are discussed. A simplified two-dimensional

analytical model is initially developed to identify the critical

parameters determining temperature distribution in the micro-

fluidic channel and justify the need for a three-dimensional model

for correct design analysis, which was developed in the commercial

software package Comsol Multiphysics 4.0. As the first step, a solid

domain modeling of the entire microfluidic chip with embedded

heaters is performed neglecting the presence of the fluid layer. The

model is used to estimate the effect of Joule heating, thermal

crosstalk in the multi-pass thermocycler chip, and quantify the

applicability of a one pass model for understanding temperature

profile in the fluid domain. The results of the solid domain simu-

lations are compared with experimental measurements of the

thermal profile in a PDMS-glass microfluidic thermocycler obtained

using a combination of thermocouples and an infrared (IR) camera.

Subsequently, the one pass numerical model examines the quality

of thermal profile in the fluid domain as a function of critical

parameters like flow rate and microchannel material. Two device

performance parameters, ramp rate, G, and maximum temperature

difference between different zones, max(DT), are defined and

evaluated with respect to variations in sample flow rate through

the device. Additionally, mesh sensitivity analysis of the simulation

models is performed to establish the numerical accuracy of the

simulations results.

2. Methods and materials

2.1. Design of the microfluidic platform

Fig. 1 defines the microfluidic continuous flow thermocycler

platform configuration used for modeling. Design of the test bed

microfluidic platform can be conceptually divided into two parts:

(1) a monolithic microfluidic chip, through which all of the bio-

logical reagents are flowed (Fig. 1a) and 2) a thin film patterned

glass wafer used to create fixed temperature distribution in space

(Fig.1b). A glass wafer (50mm (w)� 75mm (l)) patternedwith thin

film of platinum/titanium functions as a resistive heating unit, with

thermal energy dissipated from powering the heating elements

used to create the desired spatial temperature distribution. Design

of the microfluidic channels follows the basic serpentine design

proposed by Kopp et al. [18] with some modifications, discussed in

Section 2.2. PCR reagents are designed to flow through three zones,

S. Kumar et al. / International Journal of Thermal Sciences 67 (2013) 72e86 73

namely denaturation (Zone A), annealing (Zone B) and extension

(Zone C), as shown in Fig. 1a. For efficient amplification, the

serpentine configuration is implemented to pass through the zones

30 times, comparable to the 25e30 cycles used in conventional PCR

thermocyclers. Note, the microfluidic channels span (referenced

along the length of the chip, from the left edge) from w10 mm

to w65 mm (Fig. 1c). The fabrication process of the PDMS-glass

based microfluidic platform used for experimental validation of

the thermofluidic model assumption is discussed in Section 2.3.

2.2. Design of the microchannel geometry

Geometry of themicrofluidic channel is an important parameter

that needs to be understood within the context of device perfor-

mance. Typically, residence times for the denaturation and

annealing zone are lower than that for the extension zone [43]. The

residence time for the extension zone depends on the length of the

DNA segment being amplified. In this design, we consider the

residence time of PCR reagents to be in the ratio 1:1:2 for the

respective zones A, B and C. The volume flow averaged residence

time, t, in a zone is given by

ti ¼libidiQ

: (1)

where i ¼ A, B, C; l, b and d are the length, width and height of the

microfluidic channel respectively; andQ is the volumetric flow rate.

Table 1 defines the dimensions of different sections of the

microfluidic device. These dimensions give the desired ratio,

tA:tB:tCw1:1:2. The smaller cross-sectional areas of the intercon-

nect channels lowers the transition time between zones to increase

the ramp rate. The average velocity, Vavg, in the microchannels is

given by

Vavg ¼Q

bd: (2)

A significant increase in average velocity is attained through the

aforementioned change in cross-sectional area of the micro-

channels. For example, the ratio of average velocity in Zone A to the

average velocity in the interconnecting channels from Zone A to

Zone B is given by

VABavg

VAavg

¼bAdAbABdAB

¼ 18: (3)

The ratio of time spent by the fluid in Zone A to the time spent in

the interconnectingmicrochannel from Zone A to Zone B is given by

tABtA

¼lABbABdABlAbAdA

¼1

7:56: (4)

With thirty identical serpentine passes, the overall length of the

microchannel (w1.2 m) is substantial. Consequently, the hydraulic

resistance of the system must be analyzed to understand the head

loss required for device operation. The maximum allowable head

loss under which the bonding between PDMS layers remains intact,

typically around 200 kPa, limits high flow rates. Hydraulic resis-

tance of the microchannels can be estimated by using the Darcye

Weisbach formula with a Darcy friction factor, f. Hydraulic diam-

eter, Dhi, of the microchannels is used for all subsequent calcula-

tions. The head loss, hi, of a particular zone of the microfluidic

channel is given by

hi ¼ fliDhi

rfV2avg

2; (5)

where rf is the fluid density. The friction factor for laminar flow in

microchannels can be approximated as

f ¼Ci

ReDhi

; (6)

where Ci is a constant that depends on channel geometry, ReDhiis

the Reynolds number based on the hydraulic diameter [44]. In our

Table 1

Nominal dimensions of different sections of the microchannel.

Zone (i) Length of the

microchannel

(li) (mm)

Width of the

microchannel

(bi) (mm)

Height of the

microchannel

(di) (mm)

A 5.25 150 150

B 5.25 150 150

C 8 220 150

AB 12.5 25 50

BC 5 25 50

CA 5 25 50

Fig. 1. Design of the microfluidic platform (50 mm (w) � 75 mm (l) footprint): (a) flow layer; (b) glass base heating unit; (c) top view of the assembled microfluidic platform; (d)

cross-section of the microfluidic platform.

S. Kumar et al. / International Journal of Thermal Sciences 67 (2013) 72e8674

case, Ci is 57 for square channels, 59 and 62 for rectangular channels

of height to width ratio of 0.68 and 2.0 respectively. The hydraulic

diameter, Dhi, is given by

Dhi ¼4Ai

Pi¼

4bidi2ðbi þ diÞ

: (7)

From Eqs. (5)e(7)

hi ¼2mP2i li

A3i

Q ¼m2P3i li

2rfA3i

ReDhi; (8)

where m is the dynamic viscosity,

h ¼X

i

hi; (9)

where i ¼ A, B, C, AB, BC, CA and h ¼ total head loss of the device.

Assuming that the maximum allowable head loss is 200 kPa, the

maximum flow rate (analogously Reynolds number) is limited

tow 2 ml/min in PDMS basedmicrofluidic platform. Under standard

operating conditions, flow rates are on the order of 0.1e2 ml/min,

providing short loading times for sub-ml scale samples while

allowing sufficient sample residence times over the temperature

zones for complete amplification. Due to variations in cross-

sectional area along the microfluidic channel, the Reynolds (ReDh)

and Peclet (Pe) numbers are not constant. For Zone A, at flow rates

of 0.1e2 ml/min, ReDh, w0.03e0.7 and Pe ¼ ReDhPr w0.05e1.30,

where Pr is the Prandtl number. The aforementioned theoretical

calculations for Zone A were done using thermal properties of

water at 95 �C. Flows in other zones, calculated at their respective

target temperatures, show similar characteristics. As ReDhi ¼ 4rfQ/

mPi, the perimeter of the channels has a linear effect on ReDh.

Since the ReDh in Zone A is w0.1 and the Pi ratio of the smallest

cross-section channel to Zone A is w4, the flow is well within the

laminar regime. From chip design perspective the variability in

cross-section presents an interesting tradeoff between the head

loss and the reduction in transition times apart from the fabrication

challenges. A uniform cross-section will result in a lower head loss;

however, transition between zones will be longer leading to

a poorer performance of the biochemistry.

2.3. Mold and device fabrication

All microfluidic devices used in this work were prepared using

the technique of soft lithography [45e47]. All microfluidic mold

fabrication was completed in the experimental materials lab (EML)

at the MIT Microsystems Technology Lab (MTL). Photo masks were

first designed using Adobe Illustrator 11 and printed at a resolution

of 2000 dots per inch on a transparency film (CAD/Art Services Inc.,

Bandon, OR). Photolithography was used to transfer this design to

300 diameter silicon wafers to create molds for casting PDMS

microfluidic devices.

2.3.1. Mold fabrication

Silicon wafers were first placed in a Piranha solution, which is

a 1:1 mixture of concentrated sulfuric acid to 30% hydrogen

peroxide solution for about 15 min. Piranha etch cleans the organic

residues off the substrates. Wafers were then dehydrated on a hot

plate at 150 �C forw15 min.

Multilayer masters for the flow layer were fabricated using

negative photoresist SU-8 in a three-layer lithography process

[45,46]. Fig. 2a and b show the transparency masks used for

multilayer fabrication. Due to the presence of tall features, good

adhesion of SU-8 on silicon wafer is critical. First, a layer of SU-8

2002 was spun coat on a clean wafer at 3000 rpm for 60 s to coat

the wafer with a thin film of SU-8 (w2 mm), which acts as adhesion

promoter for subsequent SU-8 layers. A pre-exposure soft bake was

done on digital hotplates at 95 �C for 1 min. The entire wafer was

then exposed through broadband exposure for 40 s. This was fol-

lowed by a post-exposure hard bake for 2 min at 95 �C. The second

photoresist layer, SU-8 50, was coated on the wafer at 2150 rpm for

55 s (w50 mm nominal), followed by a soft bake (6 min at 65 �C,

20 min at 95 �C). The transparency mask shown in Fig. 2a was used

to transfer features to the SU-8 through a broadband exposure of

around 2.8 min. This was followed by a hard bake (1 min at 65 �C,

5 min at 95 �C). A third photoresist layer was then coated on the

wafer to a thickness of around 100 mm. SU-8 2050 was spun coat at

1700 rpm for 60 s followed by a soft bake (5 min at 65 �C, 20 min at

95 �C). The secondary features from transparencymask Fig. 2bwere

aligned using the alignment markers, followed by exposure of

4 min. This was followed by a hard bake (4 min at 65 �C, 10 min at

95 �C). Development was done in a single step and the unexposed

parts of SU-8 were removed by PM Acetate (1-Methoxy-2-propanol

acetate). The master molds were finally cleaned using isopropanol

and blown dry with nitrogen. Fig. 2c shows the process flow of the

steps involved.

2.3.2. Glass base heating elements fabrication

The microfluidic platform includes heating elements designed

to create the required temperature profile for PCR. The glass wafer

has patterns of platinum/titanium (Pt/Ti) thin films, which serve as

resistive heating elements. Glass wafers were first placed in

a Piranha solution for about 15 min. Wafers were then dehydrated

on a hot plate at 150 �C for about 15 min. A standard lift-off

procedure was used to deposit Pt/Ti thin film on glass wafer.

Negative photoresist NR71-3000P was spun coat on the clean glass

wafer at 3000 rpm for 40 s. This was followed by a soft bake (170 �C,

Fig. 2. (a,b) Transparency masks used in the fabrication of the multilayer mold; (c)

process flow for the fabrication of the multilayer silicon wafer mold.


4 min). A transparency mask, as shown in Fig. 3a, was used to

transfer the pattern to NR71-3000P photoresist through a broad-

band exposure of 80 s. This was followed by a hard bake (115 �C,

4 min). The unexposed parts of NR71-3000P were then removed by

developing with RD6.

Platinum has poor adhesion properties due to its noble nature;

hence a thin layer of titanium was deposited prior to platinum

deposition to improve adhesion of thin film by sputtering. Prior to

metallization, the patternedwafers were cleaned by oxygen plasma

for 30 s. First, 0.01 mmof Ti was deposited followed by deposition of

0.06 mm of Pt. This was followed by lift-off, accomplished by

immersing the wafers in RR4 developer. To accelerate the lift-off

process, RR4 was placed in a hot water bath maintained around

80 �C. The lift-off time wasw4 h. Fig. 3b shows the process flow.

2.3.3. Monolithic microfluidic chip fabrication

The microfluidic device was fabricated from PDMS silicone

elastomer (Sylgard 184, Dow Coming) using the technique of soft

lithography. Base and hardener components of the elastomer are

referred to as A and B respectively. Mixing of the PDMS components

was performed in a Thinky centrifugal mixer (Thinky USA, Laguna

Hills, CA). Consecutive replica molding frommicrofabricated silicon

masters and plasma bonding steps were used to create two-layer

elastomeric devices consisting of a layer with patterned flow

structure and a thin layer of unpatterned elastomer for capping the

chip. To facilitate the release of the elastomer during molding,

molds were first treated with perfluorooctyltrichlorosilane

(Aldrich) by placing the wafer in a large covered Petri dish con-

taining several drops of silane for 15 min.

For all the layers silicone elastomer mixture was prepared in the

following ratio; 10:1 parts A:B (w/w). After mixing, the silicone

elastomeric mixture was poured over the “flow layer” mold (4 mm

thick). A bottom channel sealing layer was created by spin coating

a thin film of PDMS on the unpatterned siliconwafer (170 rpm, 60 s)

to form a film of thicknessw 500 mm. The molds were then placed

in a vacuum chamber for 30 min to remove bubbles from the PDMS

mixture. As the microchannels are long (w1.2 m), it is imperative

that all the bubbles are removed from the PDMS mixture as even

a few bubbles lead to channel defects that promote delamination.

The degassed molds were subsequently cured for 25 min at 80 �C.

A clean razor blade was used to separate the cured elastomer

from the “flow layer” master mold. Access ports were made to the

“flow layer” using a biopsy punch (i.d. 0.5 mm) (Harris Uni-Core).

The elastomeric layer was then cleaned by first using a scotch

tape and then with acetone and isopropyl alcohol in the chemical

hood to remove the debris, followed by drying with nitrogen.

The “flow layer” with the channel side down and the PDMS

coated siliconwafer were then bonded using air plasma (500mTorr,

40 s) (Expanded Plasma Cleaner, Harrick Plasma, Ithaca, NY).

Following exposure, the “flow layer” was carefully placed against

the PDMS spin-coated silicon wafer. After bonding, the device was

placed in the oven at 60 �C for about 20min. A razor blade was then

used to separate the cured elastomer from the silicon wafer to

obtain the monolithic PDMS microfluidic chip.

2.4. Simplified two-dimensional heat transfer model

The microfluidic flow channel was initially modeled as

a simplified 2-D heat transfer model (Fig. 4), highlighting the

limitations of 2-D thermal modeling (and the need for a 3-D model

for the chip). Convection and conduction in the fluid along the

streamwise co-ordinate is considered. Conduction in fluid in y

direction is ignored as the thickness of the fluid layer is usually

small (w150 mm) and hence the thermal resistance is negligible

given the conductivity of the confined fluid (liquid) phase, kf, is

around 0.6 W/mK. By assuming heat flux, q00i ðxÞ, to be a function of

streamwise co-ordinate, one canmodel the effect of local heating in

discrete regions proximal to the zonal channels (A, B, C). Further

a heat loss factor, ai, was added which accounts for the unmodeled

heat flow in the lateral direction and through the glass substrate.

hI(x) and hII(x) are the effective heat transfer coefficients from

the top and the bottom surface of the microchannel. Note that hI(x)

and hII(x) can vary along the streamwise direction and hII(x) ¼ 0

whenever q00i ðxÞ > 0 to ensure that either heating or cooling

is modeled at the bottom of the microchannel. As a first pass

approximation, the 2-D model presented here helps to identify

critical parameters affecting temperature distribution in the

fluid. Energy balance across the control volume surface of length Dx

gives

AirfVavgCpfdTðxÞ

dx� Aikf

d2TðxÞ

dx2þ ðhIðxÞ þ hIIðxÞÞbðTðxÞ � TNÞ

¼ q00i�

x�

ð1� aiÞb;

(10)

where Cpf is the specific heat capacity of the fluid, T(x) is the local

temperature of the fluid, and Ai is the cross-sectional area of the

channel i.

The non-dimensional form of the above equation can be

written as

Fig. 3. (a) Transparency mask used in the fabrication of the glass base heating unit; (b) process flow for the fabrication of Pt/Ti thin films-based glass heating unit.


PedQ�

x*�

dx*�d2Q

�

x*�

dx*2þ

�

hI�

x*�

þ hII�

x*��

kf

D2h

d

!

Q�

x*�

¼q00i�

x*�

ð1� aiÞ

kfDTavg

D2h

d

!

;

(11)

whereDTavg is the difference between themicrofluidic chip average

temperature and ambient (TN), Q ¼ TðxÞ � TN=DTavg is the non-

dimensional temperature, d is the height of the microfluidic

channel, and x* ¼ x/d is the non-dimensional length in the flow

direction.

The 2-D model shows that the non-dimensional temperatureQðx*Þ is a function of not only microchannel dimensions (Dh,d) and

heat flux dissipated by the heaters (q00i ðxÞ), but also the heat transfer

coefficients, (hI(x), hII(x)), the heat loss factors (ai) and Peclet

number (Pe). It is important to note that hI(x) and hII(x) are lumped

parameters that account for the 3-D thermal diffusion in the

microchannel substrate and the glass wafer as well as the natural

convection to ambient. They depend on the geometry and material

of the microfluidic chip, external environment conditions and may

vary along the domain. While hI(x) and hII(x) can be fitted in a 2-D

steady state model, providing an approximation of the developed

thermal profile of the chip, the complete microfluidic chip heat

transfer problem is 3-D (with unsteady and conjugate heat transfer

occurring between the fluid and solid regions of the device when

the three thin film heaters are active).

2.5. Three-dimensional heat transfer model

The basis for the 3D simulations is as follows. Within the fluidic

domain, the energy equation neglecting the viscous dissipation

term and assuming fully developed laminar flow can be written as

rfCpfkf

�

vT

vtþ u

vT

vx

�

¼v2T

vx2þv2T

vy2þv2T

vz2; (12)

where x and u are distance and velocity in the streamwise direction,

respectively.

Note that conduction along the streamwise direction is not

ignored, as the desired temperature gradient exists along the

streamwise direction. Viscous dissipation has been neglected in the

model (per justifications in Section 3).

Inside the PDMS substrate and the glass based heat generating

unit, the energy equation takes on a simplified form, consisting of

transient and diffusion terms only, which can be written as

1

ai

vT

vt¼ V2T ; (13)

where ai is the thermal diffusivity of the material.

Different length scales are involved due to the large difference in

the size of the fluid domain (height w 150 mm, width w 150 mm)

and the solid domain (PDMS, thickness w 4 mm; glass,

thickness w 1.75 mm). The microfluidic chip consists of 30

repeating serpentine configurations (Fig. 1a). A heat transfer

simulation model through the solid domain was first developed

using Comsol Multiphysics 4.0 as shown in Fig. 5a. To model

Fig. 5. (a) Solid domain simulation model in Comsol Multiphysics 4.0; (b) meshing of the PDMS-glass simulation model.

Fig. 4. Simplified two-dimensional control volume analysis in the microfluidic

channel.


temperature distribution in the fluid domain a one pass simulation

model is established as the second step. One pass of the micro-

fluidic channel is defined as one serpentine configurationwith fluid

flow through the denaturation, annealing and extension zones

respectively as shown in Fig. 6b. The results from the solid domain

simulations are used to understand the applicability of the one-

pass model.

2.5.1. First step: solid domain model

As the solid domain simulation neglects the fluid layer,

discrepancies between modeled and experimental measurement

of the chip thermal profile are expected, principally due to two

factors: 1) the different conductivity of water and PDMS, and 2)

the convective heat transfer caused by the fluid flow inside the

channels. The change in the steady temperature profile due to the

change in thermal conductivity can be approximated by

comparing the thermal resistance of the PDMS domain with and

without the fluid layer in the vertical direction (z direction). Using

the equation diDk/(dpkp), the percentage change in the thermal

resistance is calculated to be less than 2.25%. Hence, due to

different conductivity of water and PDMS, the local temperature

profile should differ by less than w1.1 �C, 2.25% of the maximum

temperature difference in the PDMS domain (w50 �C). On the

other hand, fluid flow affects the temperature profile due to

convective transport that increases with volumetric flow rate Q.

The convective effect is expected to be cyclic in nature due to the

geometry of microchannels and heaters. Furthermore, convection

is expected to impact the temperature profile mainly in the flow

direction in the microchannel. As this change is cyclic in the

direction of the fluid flow, the overall effect of fluid flow on the

temperature distribution in the bulk PDMS domain is lower than

the local change proximal to the microchannels/fluid boundary.

Regardless of the inaccuracies induced by neglecting the heat

transfer through the fluid layer, the solid domain simulation

permits the incorporation of multiple physics of joule heating,

capturing heat diffusion through the solid domains at the device

level and clarifying the extent to which lateral heat diffusion in

transverse direction can be neglected and a one-pass model is

accurate.

To understand the effects of uneven heat generation by the

platinum heaters, due to dependence of the generated heat on

the temperature dependent electrical resistance, a coupled

Electromagnetic/Heat-Transfer (Joule Heating) simulation model

was developed comprising of the heaters, glass substrate and the

PDMS domain. A volumetric heat source (as opposed to a boundary

heat generation) is appropriate because the fraction of heat going

upward and downward is unknown. To couple the thin film domain

meshing (several nanometers thick) to the platform (few millime-

ters thick), the film was scaled up to 100 mm and the electrical and

thermal conductivity were scaled down accordingly. In correlating

real lateral heat flow with the modeled heat flow, the thermal

conductivity of the platinum layer was scaled down by 100/

0.07w1428 as qw kADT/l ¼ k0A0DT/l where A0 ¼ 1428A and k0 ¼ k/

1428. This scaling should not significantly affect the temperature

profile in the z direction as the ratio of the film thickness/platform

thickness <<1 and the effective thermal resistances are not

modified substantially (the percentage change in thermal resis-

tance in z direction is less than 2%, which produces an overall

change in the temperature of the same magnitude). The thickness

of glass domain, dg, is 1.75 mmwhile the thickness of PDMS domain

dp, is 4 mm. All top and side exposed PDMS/glass surfaces had

a natural convection boundary condition, while the lower surface of

glass had a zero heat flux boundary condition to model the Styro-

foam backing used for insulating the microfluidic platform.

Fig. 6. (a) One pass simulation model geometry in Comsol Multiphysics 4.0 and its cross-sectional view; (b) top view of the fluid domain in the one pass model; (c) meshing of the

one pass simulation model.


Tetrahedral meshing elements of different sizes (Fig. 5b) were

used to mesh the simulation domain due to large differences in the

thickness of different layers. Table 2 depicts the mesh settings. The

exclusion of the fluid domain and the geometry rescaling were

necessary to reduce the modeling complexity of the system;

nevertheless, the simulation contains the features necessary to

estimate the temperature variations through the entire micro-

fluidic platform, correct heat generation physics, realistic convec-

tive boundary conditions and correct heat diffusion through the

bulk of the domain (PDMS and glass). The mesh sensitivity analysis

is discussed in Section 2.5.3.

2.5.2. Second step: one pass model

The one-pass simulation domain for the microfluidic chip is

shown in Fig. 6a. The dimensions of the fluidic domain for the

single pass are as follows; height¼ 150 mm for the zonal channels A,

B and C and height¼ 50 mm for the interconnecting channels AB, BC

and CA; widths, bA ¼ bB ¼ 100 mm, bC ¼ 220 mm and width¼ 25 mm

for the inter-zone connecting channels; length of the serpentine

channel ¼ 4.1 cm. The PDMS domain was chosen in such a way to

encompass the fluidic domain and represent one pass of the

microfluidic device. The thickness of the top and the bottom PDMS

layers that bound the top and bottom of the fluidic domain are

(respectively) dp1 ¼ 4 mm, dp2 ¼ 0.5 mm. The simulation domain

includes glass substrate and the platinum layer to model different

volumetric heat generation by respective zones. In this work,

constant volumetric heat sources were used to realistically model

heating through thin film resistive elements. Additionally, in the

model, the thickness of the platinum heater (0.07 mm) was scaled

up to 100 mm for ease of coupling meshing of the thin film domain

with the rest of the domain and as discussed earlier the thermal

conductivity was scaled appropriately.

The total length (dimension along the transverse direction) of

the one-pass model is 1.815 mm and the width (dimension along

the flow axis) is 25.5 mm. Out of the 25.5 mm, 21.5 mm is the true

length of the section extracted from the microfluidic platform. The

remaining 4 mm consists of two 2 mm segments of sidewall

materials (Fig. 6a), which model the lateral heat flows (along the

flow axis) through the 18.5 mm of PDMS and 28.5 of glass to

ambient. This length scaling was similarly accompanied by the

scaling of the sidewall material’s thermal properties. To simulate

lateral diffusion of heat to ambient through 14.25 mm thickness of

glass (on either side) and 9.25 mm of PDMS through the 2 mm of

material on the side walls, the thermal conductivity of the addi-

tional sidewall material were scaled down by 7.125 (for glass) and

4.625 (for PDMS) as q w kADT/l ¼ k0ADT/l0 where k0 ¼ k/7.125 and

l0 ¼ 2l/14.25. The thickness of glass domain, dg, is 1.75 mm.

A free tetrahedral meshing scheme was used to mesh the

domain (Fig. 6c). Though all domains have rectangular geometry,

the solid and the fluid domains have different length scales and

there are significant variations in cross-section in the fluid domain

that makes rectangular mesh a difficult option. Comsol permits

adaptive meshing in different domains, achieving acceptable mesh

quality in all the domains. The mesh settings are shown in Table 2.

Thermal boundary conditions were defined as follows: 1)

natural convective heat transfer coefficient on the top PDMS

surface exposed to air and on the side PDMS and glass surfaces

exposed to air; 2) periodic heat condition on the side walls of PDMS

and glass which form the cutting plane along which the simulation

domain is separated from the actual device (Fig. 6a Cutting plane 1,

Cutting plane 2), 3) zero heat flux boundary condition on the lower

surface of the glass substrate as the microfluidic platform was

insulated by a Styrofoam backing.

The fluid properties were modeled as water, using Comsol to

incorporate the temperature dependent variations of the thermal

properties. As dilute solutions of biological reagents are considered

(mM to mM range), the variations in thermal properties due to

presence of reagents can be neglected. Hence, pure water was used

to approximate the thermal properties of the aqueous solution. The

constant thermal properties of the PDMS and glass used are;

kp ¼ 0.15 W/mK, ap ¼ 9.34 � 10�8 m2/s, kg ¼ 1.38 W/mK,

ag ¼ 7.81 � 10�7 m2/s.

2.5.3. Mesh sensitivity analysis

In both the glass-PDMS and one-pass simulation models,

meshing granularity was chosen to have an adequate number of

mesh elements with good mesh quality while maintaining

a reasonable computational complexity. The mesh sensitivity

analysis was performed for the glass-PDMS heat transfer model

(Fig. 5) by varying the number of meshing elements and repeating

the simulations with themodified setup. In the glass-PDMS heating

model, there are two meshing subdomains: A) the thin film heater

and B) the PDMS and the glass substrate. For each domain, the

meshing density was decreased independently, i.e. the meshing

settings for one domain were kept constant while they were varied

for the other. A specific point in the domain was chosen for

studyingmesh sensitivity; x¼ 37.5 mm, y¼ 17.5 mm and z¼ 0mm.

Fig. 7a and b show the absolute error of the temperature obtained

at the specific point with respect to the temperature obtained with

the maximum number of elements (Tref) by varying the meshing

settings of the two meshing domains A and B. As the mesh setting

of one meshing domain was changed, the number of meshing

elements in the other meshing domain changed slightly, which is

indicated by the number next to the data point in Fig. 7a and b. It

was observed that the absolute error is less than 0.15 �C due to

variations in themeshing density. The temperature variation due to

meshing is smaller than the error induced by discrepancies

between the model and the real device. We observe that the slope

of the error curve decreases sharply with increase in the number of

elements; further refinement will have a smaller marginal

improvement in the numerical accuracy of themodel. The results in

the paper are presented from simulations carried out with the

maximum mesh density noted in Fig. 7.

2.6. Experimental procedure

Experimental validation of the 3D thermofluidic model was

performed with a PDMS/glass continuous flow microfluidic chip

(described in Section 2.1) via infrared thermography and thermo-

couple temperature measurements. To operate the experimental

platform, the PDMS chip was placed on top of the heat generating

glass unit and alligator clips were used as electrical connects. PCR

solutionwas flowed through themicrofluidic channels via a syringe

pump. The Peclet number of the flow varied in the range 0.07e0.7.

Table 2

Meshing settings for simulations.

Meshing type: tetrahedral

Domain Comsol setting No. of

elements

Volume

(mm3)

Average

quality

A. Joule heating simulation

Thin film Coarse 42,747 56.25 0.5573

Glass Fine and normal 191,660 6506 0.5306

PDMS Normal 131,910 9600 0.5921

B. One pass simulation

Interconnecting channels Extremely fine 37,280 0.02813 0.8302

Zonal channels Extremely fine 9116 0.5002 0.825

Heaters Finer 2739 1.361 0.7043

Rest of the domain Finer 429838 264.37 0.7958


Higher flow rates were limited by the delamination of the PDMSe

PDMS bonding.

Thermocouples were placed every 7.5 mm along each heating

zone on the undersurface of the glass wafer (the only surface free to

attach thermocouples) for real time temperature monitoring. To

correlate the temperature on the undersurface of the glass with the

fluid within the microchannels, the temperature difference across

the intermediate PDMS and glass heater was estimated. The

temperature difference between the undersurface of the glass

wafer and the microchannels was estimated in two steps: 1) the DT

between the top and bottom of the glass-heater structure was

initially measured using thermocouples, and 2) the temperature

difference between the top surface of the glass heating unit and the

fluid channels, separated by a thin PDMS layer, was calculated. It

was experimentally observed that the bottom glass surface was

cooler than the top by DT w 2 �C. This temperature difference is

comparable to the temperature difference across the PDMS layer

bounding the fluid channels above the heaters on the topside of the

glass. The area-specific thermal resistance, Rthermal, of the PDMS can

be approximated as

Rthermal ¼dmkmw3:33� 10�3 m2K=W; (14)

where dm is the thickness of thematerial, and km is the conductivity

of the material. The temperature difference, using zone A, which

has the highest q00

, as an upper bound, can be estimated as:

DT ¼ q00Rthermalw3 �C: (15)

Since the temperature drop from the heater at the glass top

surface to the glass bottom surface and the temperature drop from

the heater to the channels inside the PDMS are practically the same,

the temperature on the underside of the glass can be approximated

to be the temperature of the fluid inside the channels withinw1 �C.

3. Results and discussion

3.1. Evaluation of thermal crosstalk from solid domain simulation

and experimental measurements

A coupled joule heating and heat transfer glass-PDMS simula-

tion was first used to quantify the validity of the one-pass model.

The following values of input voltages gave the temperature

distribution consistent with our target zonal temperature

requirement: VA ¼ 10.2 V; VB ¼ 3.6 V; VC ¼ 2.4 V. The input voltage

required for Zone C, maintained at 71e75 �C, was found to be less

than the heat flux required for Zone Bmaintained at 58e62 �C. This

observation can be explained by the considerable y direction lateral

heat diffusion in the glass substrate due to the high thermal

conductivity of the glass wafer.

The joule heating model provides an estimate of the uniformity

of heat generated per unit volume. Fig. 8a and b show that the heat

generated per unit volume varies by w3% along the direction of

the heater (transverse direction). Furthermore, in the one-pass

model, the largest source of error is in approximating the lateral

thermal diffusion in x direction (transverse direction) by periodic

thermal boundary conditions. Fig. 9a and b show that most of the

heat in the transverse direction flows via the glass domain and not

Fig. 8. (a,b) Variation in volumetric heat generation along the transverse direction.

Fig. 7. Variation in the absolute error of temperature at a specific location with the

variation in meshing density of a) thin film heaters b) glass and PDMS substrate. The

absolute error is less than 0.15 �C and the slope of the error curve decreases sharply

with increase in meshing density implying smaller marginal improvement in

numerical accuracy with further mesh refinement. Note that the numbers next to the

data points indicates the number of mesh elements in a) glass and PDMS substrate and

b) thin film heaters.


through the PDMS. In the central 80% of the microfluidic chip, the

heat flux in the x direction grows linearly from the center to the

end. It was observed that the variation is from 0 to w100 W/m2

through the PDMS domain and 0ew900 W/m2 through the glass

domain. More importantly, for the 1.815 mm wide one pass

model control volume, the net x direction heat flow should be

negligible compared to the z direction heat flow added by

the heaters. In the central 80% of the PDMS domain 500 mm

above the thin film heater, the transverse direction heat flux can

be approximated as q00A;x ¼ 4800(x � 0.0375) W/m2,

q00B;x ¼ 3600(x � 0.0375) W/m2 and q00C;x ¼ 4200(x � 0.0375) W/

m2; where x ¼ 0.0375 m is the center of the chip. For

a 1.815 mm thick one pass cycle, the net heat loss in the x

direction is q00A loss;x w 8.7 W/m2, q00B loss;xw 6.5 W/m2,

q00C loss;xw7.6 W/m2. The following are the z direction heat flow

into PDMS: q00A;z ¼ 1500 W/m2; q00B;z ¼ 400 W/m2; q00C;z ¼ 380 W/

m2. Hence the total heat loss in the x direction can be neglected

compared to the heat added by the heaters in z direction;

q00loss;xAx=q00zAz << 1, since Ax/Az ¼ 4/1.8. Furthermore, it was

observed that the y direction (flow axis) heat flux has sharp

gradients along the flow axis, highlighting the variability of flow

axis heat flux and need of a three-dimensional model to capture

this significant mode of lateral diffusion.

As the magnitude of the heat flow in the transverse direction

grows with distance from the center, it forces a temperature drop

away from the center of the microfluidic chip. Fig. 10a shows the

simulated temperature variations along the transverse directions in

the PDMS domain. Around two-third of the PDMS domain satisfy

4 �C variability from the maximum temperature.

Fig. 10a shows a comparison of the simulated vs. mean experi-

mental steady state temperatures along the transverse direction for

the three zones. The transverse temperature gradients, as expected,

are relatively flat (�4 �C variability) across themajority of the zones

(from 20 mm to 55 mm), becoming more pronounced at the edges.

Two factors contribute to the edge effects. First is the nature of

thermal diffusion with a constant heat generation and convective

heat transfer as boundary condition. Second, the electrical connects

used to power the heating element has a finite thermal resistance;

hence, it conducts heat and enhances the drop in temperature at

Fig. 9. (a) Heat flux in transverse direction inside the glass domain on a plane 500 mm

below the glass-PDMS interface; (b) heat flux in transverse direction inside the PDMS

domain on a plane 500 mm above the glass-PDMS interface. Note that the figure bars

are in W/m2.

Fig. 10. (a) Variation of temperature in transverse direction obtained through simu-

lation of the PDMS-glass domain and experimental measurements on glass. The

“Simulation-glass” are temperatures obtained from simulation on the underside of the

glass, the “Simulation-PDMS” are temperatures in PDMS on a plane 500 mm above the

glass-PDMS interface obtained from simulation, the “Experimental-glass” are

temperatures obtained from thermocouple measurements on the underside of the

glass. The simulated heat input values are 1.524 W, 0.20 W and 0. 09 W, in zones A, B

and C respectively. The experimental heat dissipated values are 2.13 W, 0.39 W and

0.19 W, in zones A, B and C respectively. The vertical lines indicate the span of the

microfluidic channel which is from w10 mm to 65 mm; (b) simulated steady state

temperature distribution in the PDMS-glass domain.


the ends. Fig.11 shows infrared images of the devicewhen the three

thin film heaters are operating, and is in good agreement with the

simulated results of Fig. 10b. The IR images substantiate the

assumption that the gradients in transverse direction are negligible

vs. the flow axis direction. Note that the infrared images provide

information about the temperature of the top PDMS surface rather

than the temperature on the glass surface.

Achieving steady state thermal equilibrium within a micro-

fluidic chip is hardly instantaneous, and needs to be considered

when using a material like PDMS as a substrate. PDMS is a silicon-

based organic polymer, and, due to its low thermal diffusivity

(apw10�7 m2/s), the time constant for temperature diffusion is

high. It is difficult to obtain exact analytical solutions to general

multidimensional transient heat conduction problems. Typically,

a lumped system model neglecting conduction in the domain can

be employed to obtain an approximation to the time to reach

steady state. For such a model to be valid, one requires Biot number

(Bi) � 0.1. From the steady state solid domain simulation, the

average natural heat transfer coefficient for the top surface of the

PDMS was found to bew10 W/m2K. The Bi for the PDMS was then

calculated to be 0.267 and the time to reach 99% of the final steady

state, s, on the order of 3 � 103 s. Strictly speaking, the aforemen-

tioned calculation is not applicable to our system not only due to

the high Bi but also due to variable natural heat transfer coefficient

on the exposed surfaces, nevertheless it can be used as a rough

estimate. Note that the time constant of the transient process is

inversely proportional to Fourier number (Fo) � a function of Bi,

depending on the value of Bi and the boundary conditions [48]. For

lower waiting times, a higher Fo is desirable and can be obtained

through selecting a higher thermal diffusivity material.

Unsteady simulation of the glass-PDMSmodel was performed in

Comsol with the option of generalized-alpha transient solver and

automatic time stepping was used with the initial trial step of

0.001 s. The simulation provides an approximation to s, on the

order of 2 � 103 s. The high value of s has practical implications in

terms of estimating the wait time necessary for the device to reach

a steady state thermal profile, particularly when applied to the

engineering of “rapid” field-based diagnostic chips. Operating the

device in the transient state, prior to adequate warm-up, can

detrimentally affect the yield and quality of the resultant PCR

products, or inhibit the PCR amplification entirely. Simulation

predicted the time required to warm-up the device to final steady

state temperature to be w30 min. To minimize this time when

performing the experiments, the voltage of the warmer zone, zone

A, was initially ramped up to w1.5 times the value applied under

steady state operation, reducing the device warm-up time

to w10 min.

3.2. One pass simulation results and device thermal performance

The one-pass thermofluidic numerical model developed in this

manuscript was used to evaluate the effect of two parameters,

volumetric flow rate and microfluidic chip material, on the steady

state thermal profiles of continuous flow PCR devices. The

numerical model has the capability to comprehensively capture

conjugate thermal transport processes and thermal crosstalk due to

lateral diffusion and convection, making it useful as a design tool in

the layout of microfluidic fluidic circuitry, as well as the integration

of external heating elements with microfluidic chips.

The thermofluidic model was initially applied to a PDMS-glass

based microfluidic platform, evaluating the steady state chip

thermal profile as a function of linear flow rate through the serpen-

tinemicrochannel. The volumetric heat generation values, which are

the parameters of simulations, were varied in order to achieve the

desired temperature distribution. The following values of volumetric

heat generation gave the temperature distribution consistent with

our target zonal temperature requirement (60 �C e anneal, 73 �C e

extension, 92 �C e denaturation); q000

A ¼ 6:3� 107 W/m3,

q000

B ¼ 3:9� 104 W/m3, q000

C ¼ 2:4� 105 W/m3. For the PDMS-glass

microfluidic platform, steady state temperature profiles in the fluid,

glass andPDMSdomain are shown inFig.12a forVavg¼0.5mm/s. The

contribution of viscous dissipation was ignored in the above

modeling. The average heat perunit volume, q000 generated by viscous

dissipation can be estimated as q000

¼ mðvu=vyÞavg, where m is the

viscosity and (vu/vy)avg is the average velocity gradient in the

channel. (vu/vy)avg can be approximated as Vavg/Dh, where Vavg is the

average velocity, and Dh is the hydraulic diameter of the channel. A

worst-case analysis can be donewith high flow rate ofQw 2 ml/min,

Fig. 11. Infrared images of the microfluidic platform at different times. The glass temperatures indicated are measured by thermocouples placed on the undersurface of glass. The

total power consumed during operation of the microfluidic chip is w3 W. The figure bars are in �C.


considering the interconnecting channels with a minimum

Dh ¼ 33 � 10�6 m, while approximating q000

wmQ=D3h. Using Q ¼ 2 ml/

min, m ¼ 0.001 kg/ms and Dh ¼ 33 � 10�6 m, q000 w 0.9 W/m3. By

inspection, comparing the contribution of heating by viscous dissi-

pation (0.9W/m3) to that of electrical heating,>104W/m3, the effect

of viscous dissipation is negligible.

The centerline residence time from numerical simulations was

calculated as the ratio of streamwisedistance to centerlinevelocity in

a particular zone. From the simulations, we obtain; tAw9.5 s, tBw8 s,

tCw30 s forQ¼ 0.675 ml/min (corresponding toVavg¼ 0.5mm/s). The

flow in the microchannel is laminar and hence exhibits a parabolic

velocity profile where the velocity at the centerline is highest and is

zero at the wall due to the no-slip boundary condition. Hence the

minimumresidence timeoccurs at the centerline. The residence time

increases with distance from the center and reaches infinity at the

wall. In order to reduce the variability in residence time, flow

focusing, a technique that confines reactants to the center of the

microchannel by injection of additional buffer, could be imple-

mented on chip as it has been done in other applications, e.g. cell

sorting and droplet generation [49]. Nevertheless, flow focusing is

limited to devices in which reactants diffusion has minimal effects,

i.e. as the chemicals flow they remain centered within the micro-

channel, i.e.minn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dl=Vavg;i

q oD

minn

bi=2; di=2o

, where D is the

diffusivity of the reacting chemicals, l is the total length of the

microfluidic channel and Vavg,i is the average velocity for each of the

zone [50]. For the device simulated in this paper, considering the

maximum average velocity and hence minimum diffusion, the

effective diffusion length for a 100 base pair DNA molecule is

w22 mm, a length larger than half the minimum channel size,

12.5 mm, which prevents the effective implementation of flow

focusing to reduce residence time variability.

The ramp rate of a zone, G, was calculated as the average rate of

change of temperature moving from that zone to the next zone

Gij ¼Ti � Tjti � tj

; (16)

where the subscript, i, j, denotes either Zone A (i ¼ A, j ¼ A), Zone B

(i ¼ B, j ¼ B) or Zone C (i ¼ C, j ¼ C). For Vavg ¼ 0.5 mm/s, the ramp

rates were calculated to be GABw� 7:5 �C/s, GBCw1:6 �C/s, andGCAw16:5 �C/s.

To understand the effect of volumetric flow rates on tempera-

ture distribution, simulations were carried out by varying the inlet

velocity, 0.5 mm/s � Vavg � 5 mm/s, while keeping the dissipated

heat flux values constant. Variation of volumetric flow rate results

in variation of Peclet number, Pe, which is the dimensionless ratio

of convective to conductive heat transport. Fig. 12b shows the

variation of temperature along the streamwise direction at the

centerline of the microchannel. Convective effects increase with an

increase in volumetric flow rate resulting in a flatter temperature

profile across the denaturation, annealing, and extension zones.

Ultimately, at high Pe, PCR failure can be attributed to both the

degradation of the temperature profiles within the zones and the

short residence time in the extension zone, which, consequently,

does not allow sufficient time for product extension by the poly-

merase. Note that Pe ¼ ReDhPr. Typically microfluidic devices for

PCR like applications involve aqueous solutions with Pr similar to

that of water. Another approach ofmodulating the convective effect

would be to modify Pr of the solution, as permissible by the

biochemistry, hence inducing changes in the temperature profile.

From a design perspective for a microfluidic PCR device, a crit-

ical parameter is the temperature variation that a fluid particle

experiences as it passes through the microfluidic channel. The

material derivative of temperature at steady state can be written as

DT

Dt¼ u

vT

vx: (17)

As we are considering fully developed flow at low Re, the

advective term has a contribution only from the streamwise

velocity, u. It is important to note that for joule heating-based thin

film heaters, a constant volumetric heat generation boundary

condition was used as opposed to constant temperature. Fig. 13

shows the steady state variation in temperature profile of the

fluid at the centerline of the channel with time as it flows through

the channel for Pe ¼ 0.3375.

Comparing the continuous flow microfluidic thermal profile to

an ideal stepped thermocycling profile, the trade off between

sample throughput and residence time in the target zones becomes

evident. Increasing sample throughput by increasing the flow rate

through the device affects both the ramp rate, G, and the maximum

temperature difference which can bemaintained between different

zones, max(DT). Fig. 14 shows the variation of the device perfor-

mance parameters (abs(GAB), the ramp rate of Zone A (other ramp

rates will show similar behavior), and max(DT)) as a function of Pe.

Fig. 12. (a) Temperature distribution in the one pass model for Pe ¼ 0.3375; (b)

variation of temperature profile along the streamwise position in the microfluidic

channel vs. Peclet number. Volumetric heat generation was maintained constant

throughout the simulations. A, B and C define the different temperature zones in the

microfluidic channel. Low Pe corresponds to Pe ¼ 0.3375, 0.675, 1.35. High Pe corre-

sponds to Pe ¼ 2.025, 2.7, 3.375.


As Pe increases, the absolute value of ramp rate between zones

increases, but at the expense of max(DT), which compromises PCR

efficiency.

Channel geometry and length also affect the limits on flow rates.

Channel cross-section presents an interesting trade off between the

head loss and the transition times or sharpness of the temperature

profile. A sufficient number of cycles are required for adequate

biochemical amplification and, as discussed in Section 3.1, the end

effects that can adversely deteriorate temperature profile inw35%

of the chip have to be accounted for when designing the length of

the microfluidic channel. A possible approach to reduce the length

of the microchannels would be to reduce the spacing between the

heaters, but this comes at the cost of increasing the thermal

crosstalk between the zones. Note that 63% of the total heat is

dissipated in Zone A, while the heat dissipated in the other two

regions act primarily to fine tune temperature distributions tomeet

the zonal requirements (Section 3.1).

Apart from the Peclet number, another important parameter

governing temperature distribution in the fluid domain is the

thermal conductivity of the bounding microfluidic device material.

Continuous flow PCR devices fabricated from PDMS, glass and

polystyrene, for example, have significantly different steady state

thermal profiles, as the heat transport mechanism is significantly

affected by thermal diffusion through the microchannel materials.

Fig. 15 shows a comparison of the variation of steady state

temperature profiles in the fluid domain along the streamwise

position for continuous flow PCR microchannels in the aforemen-

tioned materials (all bound by a bottom glass surface). The value of

thermal conductivity used for different materials are kp(PDMS) ¼ 0.15 W/mK, kg (glass) ¼ 1.38 W/mK and kps(polystyrene) ¼ 0.08 W/mK. An inspection of the temperature

profiles for the three materials shows that, as expected, the mate-

rials with lower thermal conductivity (PDMS, polystyrene) have

more defined temperature profiles, while, with high thermal

conductivity materials such as glass, rapid diffusion of heat through

the substrate has the undesirable effect of flattening the thermal

profile.

3.3. Computational cost of simulations

Finally the computational costs and advantages of the one pass

simulation technique were evaluated when compared to the ther-

mofluidic modeling of the entire microfluidic chip. The default

generalized minimal residual method solver (GMRES) and the

flexible GMRES (FGMRES) were used to solve the one pass and the

solid domain model respectively. GMRES is an iterative method to

solve a system of differential equations and yields convergence

after O(n) operations when the solution of the system of differential

equations yields a sparse matrix and O(n2) when the matrix is

dense, where n is the number of degrees of freedom [51,52].

FGMRES differs marginally from GMRES as it allows for the intro-

duction of known vectors in the solution search space, which can

potentially reduce the computational cost.

In order to study the variation of solution time with mesh

resolution the number of elements for each step, the solid domain

and the one pass simulation, were varied and the solution timewas

recorded. Varying the number of elements in the finite element

model resulted in the variation of the number of degrees of

Fig. 13. Temperature time history of fluid particles at the centerline of the microfluidic

channel obtained through simulation for Pe ¼ 0.3375. Comparison with an ideal

thermocycling profile is presented.

Fig. 14. Variation of device performance parameters (abs(GAB) and max(DT)) with

Peclet number (Pe). Low Pe is suitable for device operation. At high Pe, the max(DT)

(between the denaturation and annealing zones) is inadequate, promoting non-

specific PCR product formation and reaction failure.

Fig. 15. Variation of temperature profile along the streamwise position in the micro-

fluidic channel for different microchannel material. Heat flux input and the volumetric

flow rate (or Pe) were maintained constant throughout the simulations.


freedom. In both the cases it was observed that as the total number

of degrees of freedom is increased, the solution time increases

linearly, which is the characteristic of a sparse system (Fig. 16). In

the case of the solid domainmodel a large proportion of the volume

is subjected only to thermal diffusion, which explains the sparsity

of the resulting matrix and makes the problem numerically easier

to solve. In the case of the one pass simulation, the solution of the

NaviereStokes equations coupled with the heat transfer is typically

more expensive than solving Joule heating and temperature diffu-

sion problems. Nevertheless, the simulation exhibits a linear

dependence on n, which indicates a system associated with

a sparse matrix. In this case the high proportion of solid domain

compared to the fluid domain can provide an explanation for the

linear dependence.

In Section 2.5.3, we noted that the numerical accuracy increases

non-linearly with the number of elements with a small marginal

improvement beyond a certain threshold. The linear increase in

computational time hence calls for a careful decision when

selecting the mesh resolution to balance the tradeoff between

numerical accuracy and computational cost. The computation

advantage of the one-pass approach is highlighted by the linear

increase in computational time with the number of degrees of

freedom. The computer time needed for the one pass simulation

would be around 1/m the one needed for a full geometry simula-

tion, where m is the number of heating cycles (m ¼ 30).

4. Conclusions

Microfluidics is emerging as an exciting technology to perform

chemical and biological experiments at low cost and high

throughput. An important requirement for widespread acceptance

of this technology is comprehensive design analysis and subse-

quent optimization. As microfluidic chips become more complex,

with integrated electrical components, including heating elements,

thermal chip modeling becomes an increasingly important part of

the design process. In this manuscript, a computationally efficient,

two-step modeling methodology to study continuous flow micro-

fluidic thermocyclers was presented. Detailed thermal perfor-

mance of the chip was studied, like quantifying thermal cross talk,

time required to reach steady state and thermal profile in the

microchannel with variation in volumetric flow rate (through

Peclet number) and microchannel material. It was found that the

design studied here meets the thermocycling requirement at low

flow rates though high flow rates lead to deterioration in thermal

profile. A different design will exhibit different variability in fluid

domain thermal profile with operating conditions due to the

complex three-dimensional thermal coupling between the fluid

and solid domain. The use of a three-dimensional thermofluidic

model as described in this paper ultimately enables the micro-

fluidic designer to identify potential limitations of chips that

include heat dissipating elements, such as thermal crosstalk

between microchannels, study design optimization by varying

parameters like geometry and material and incorporate manage-

ment solutions, rather than cycle through iterative prototypes. The

authors anticipate that the methodology presented in the current

work will serve as valuable tools in physical microfluidic device

design process, reducing the time required to fabricate functional

prototypes while maximizing reliability and robustness.

Acknowledgments

The authors will like to thank the following funding agencies: SK

was supported by the Singapore-MIT Alliance (SMA) and MAC was

supported by CONACYT, Mexican National Science and Technology

Council (Grant 205899).

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Nomenclature

The subscript j or i refer to different zones of the microfluidic channel, which are Zones

A, B, C or the interconnecting channels AB, BC, CA. Units used are indicated betweenbrackets, [ ], unless otherwise indicated in the text or corresponding equation.

A: cross-section area [m2]Ai: cross-section area of the i section of the microchannel [m2]Bi ¼ hndm/km: Biot Numberb: width of the microfluidic channel [mm]bi: width of the microfluidic channel in a particular zone i [mm]Cpf: specific heat capacity of the fluid [J/kgK]D: DNA diffusivity [m2/s]Dh ¼ 4A/P: hydraulic diameter [m]Dhi: hydraulic diameter in a particular zone i [m]dm: thickness of the material layer m, where m ¼ g (glass), p (PDMS) [m]Fo ¼ amt=d

2m: Fourier Number

f: Darcy friction factorg: acceleration due to gravity [m/s2]h: head loss [Pa]hi: head loss in zone i [Pa]hn: natural heat transfer coefficient [W/m2K]hI: effective heat transfer coefficient from the top surface of the microchannel

[W/m2K]hII: effective heat transfer coefficient from the bottom surface of the microchannel

[W/m2K]km: thermal conductivity of material m, where m ¼ f (fluid), g (glass), p (PDMS), ps

(polystyrene) [W/mK]l: total length of the microfluidic channel [m]li: length of the microfluidic channel in a particular zone i [m]max(DT): maximum temperature difference which can be maintained between

different zones [�C]P: perimeter [m]Pi: perimeter of zone i of the microchannel [m]Pe: Peclet numberPr: Prandtl numbersQ: volumetric flow rate [m3/s]q00i;k: heat flux dissipated in a particular zone i in direction k; k ¼ x, y or z [W/m2]

q000: heat generated per unit volume [W/m3]q

000

i : heat generated per unit volume in zone i [W/m3]Rthermal: thermal resistance [W/mK]ReDh ¼ rfQDh/Am ¼ 4rfQ/Pm: Reynolds number based on hydraulic diameterT: temperature [�C]Tref: reference temperature calculated at the finest mesh resolution [�C]TN: ambient temperature [�C]t: time [s]ti: residence time in a particular zone i [s]u: streamwise velocity [m/s]Vavg: average velocity [m/s]V iavg : average velocity in a particular zone i [m/s]

x: stream wise co-ordinate [m]x* ¼ x/Dh: non-dimensional stream wise co-ordinatey: co-ordinate perpendicular to the streamwise direction [m]z: co-ordinate perpendicular to the streamwise direction [m]am: thermal diffusivity of the material m, where m ¼ f (fluid), g (glass), p (PDMS), ps

(polystyrene) [m2/s]b: coefficient of thermal expansion of air [1/K]V2: Laplacian [1/m2]

Dk: kf�kp [W/mK]DT: temperature difference [�C]Q ¼ TðxÞ � TN=DTavg : non-dimensional temperatured: height of the microfluidic channel [mm]di: height of the microfluidic channel in a particular zone i [mm]dg: thickness of the glass domain [mm]dp1: thickness of the top PDMS layer [mm]dp2: thickness of the bottom PDMS layer between the fluid and the glass [mm]rm: density of the material m, where m ¼ f (fluid), g (glass), p (PDMS), ps (poly-

styrene) [kg/m3]Gij: ramp rate of zone i where j is the zone following zone i [K/s]s: time to reach 99% of the final steady state [s]m: dynamic viscosity [kg/ms]


Thermal modeling and design analysis of a continuous flow microfluidic chip

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