Numerical analysis of transport processes in a microfluidic device for bacteria confinement

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CONFIDENTIAL DOCUMENT. ONLY BE USED FOR PURPOSES OF EVALUATION

Numerical analysis of transport processes in a

microfluidic device for bacteria confinement

Master Thesis

by

Joana Maria Pereira Ferreira

Developed within the Dissertation course

held in the

Simulation Technologies Laboratory (SIMTECH Lab)

Transport Phenomena Research Centre (CEFT/FEUP)

Supervisor: Prof. João Moreira de Campos Co-supervisor: Dr. Tiago Sotto Mayor

Department of Chemical Engineering

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Acknowledgements

First and foremost, my deepest gratitude goes to my supervisors. To Professor João Campos, for the opportunity, guidance, and invaluable support. To Doctor Tiago Sotto Mayor, for all the meetings and discussions that pointed me in the right direction.

I also wish to thank the many colleagues I have met in the CEFT group, for the friendly work environment provided.

I am grateful to my family and friends, especially my parents, my brother, and my grandparents for their dedication and encouragement.

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We cannot solve our problems with the same thinking we used when we created them. Albert Einstein

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Resumo

No início da década de 1980 surgiu a microfluidica que, embora numa fase de pesquisa exploratória, apresenta uma elevada margem de progresso. Deste modo, existem aplicações práticas onde as vantagens da microfluidica se destacam claramente em oposição às da macroescala convencional, realizando-se procedimentos experimentais impossíveis à macroescala. Um elevado espetro de aplicações inclui o campo de diagnóstico (medicina), síntese química (química), separação celular e replicação do ADN (biologia), bem como outras áreas distintas, onde se insere a indústria aeroespacial, a engenharia de polímeros, e os dispositivos eletrónicos e fabrico de biossensores.

Nos microdispositivos, a investigação numérica do escoamento e transporte de massa em escoamento líquido monofásico tem sido desenvolvida durante as duas últimas décadas. Nesta tese procurou-se investigar o escoamento e transporte de massa através de um dispositivo à microescala para captura de bactérias. Para tal, usou-se a dinâmica de fluidos computacional para investigar as condições que afetam o metabolismo das bactérias (representado por uma reação à superfície destas) – fornecimento contínuo de nutrientes e remoção dos produtos resultantes do metabolismo -, assumindo o crescimento bacteriano em monocamada. Um modelo computacional em 2D do dispositivo foi desenvolvido e os principais fenómenos de transporte foram validados comparando com dados experimentais e numéricos provenientes da literatura.

Os resultados obtidos indicam concluir que os parâmetros que mais afetam a reação de glicólise à superfície das bactérias são: cinética reacional (estequiometria da reação e modelo cinético), comprimento e forma das bactérias, e comprimento dos microcanais em que ocorre captura/crescimento das bactérias. A reação de glicólise é favorecida, sob o ponto de vista do aumento da taxa de formação de ácido láctico à superfície das bactérias e maior consumo da glicólise existente, em microcanais longos e bactérias alongadas com uma parte fora dos microcanais, modelos de reação de primeira ordem, e bactérias em forma de bastonete, em alternativa à forma esférica.

Palavras-chave: microfluidica; dinâmica de fluidos computacional; reação à

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Abstract

Microfluidics emerged in the beginning of the 1980’s and it is still in a state of exploratory research, but with a high growth margin. For that reason, there are practical applications where its advantages stand out clearly over conventional macroscale, performing experimental procedures impossible at the macroscale. A broad spectrum of applications includes diagnostic field (medicine), chemical synthesis (chemistry), cell separation and DNA amplification (biology), as well as other different areas, including aerospace industry, polymer engineering, and electronic devices and biosensor fabrication.

Numerical investigation of fluid flow and mass transport in single-phase liquid flow in microfluidic devices has been carried out during the past two decades. In this thesis, one aimed at investigating the fluid flow and mass transport through a microdevice for trapping of bacteria. For that purpose, computational fluid dynamics (CFD) was used to investigate the conditions affecting the bacterial metabolic process (represented by a reaction at the bacteria surface) – continuous supply of nutrients and removal of metabolic products -, with bacterial growth in a monolayer. A 2D numerical model of the device was developed and the main transport phenomena were validated using experimental and numerical data from literature.

The obtained results show that the parameters influencing the glycolysis reaction at bacteria surface are: reaction kinetics (stoichiometric reaction and kinetic model), bacterial length, bacteria shape, and length of the microchannels where bacteria trapping/growth occurs. Higher lactic acid production at the bacteria surface and higher glucose consumption is promoted, in longer microchannels and elongated bacteria with a portion outside of the microchannel, first order reaction models, and rod-shaped bacteria, instead of spherical shape.

Key-words: microfluidics; computational fluid dynamics; surface reaction;

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Statement

I hereby declare that this is an original work and that all non-original contributions needed to its completion were dully referenced with the respective source clearly identified.

(Joana Ferreira) July 2016

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List of Figures

Figure 2.1 – Schematic representation of size comparison of biological targets and microfluidic technology (Adapted from 19_{). ... 4}

Figure 2.2 - (a) Schematic representation of the PDMS device with bound cover glass (Adapted from 20_{), (b) Top view}

of a microdevice example, and (c) 3D conformation, with the indication of main channel and microchannels, and the respectively inlet and outlet (Adapted from 21_{). ... 5}

Figure 2.3 - Examples of cell trapping strategies in microfluidic devices. (a) Picolitre bioreactor for cultivation of bacteria, when the reactor is fully packed cells are pushed out of the overflow channels; design allows for continuous cultivation and analysis (Adapted from 22_{). (b) Optical trapping in a photo-polymeric hydrogel (microfluidic device}

with three separate inlets; the red vertical arrow indicates the chemoeffector gradient in the direction of the highest concentration) 23_{. (c) Hydrodynamic trapping using sieves (U-shaped with an aperture); streamlines are also}

represented, as well the red stagnation points (Adapted from 24_{). (d) Trapping bacteria through the optical force}

exerted on them along a microfluidic channel 25_{. ... 6}

Figure 2.4 - Molecules in a parallel-plates channel. (a) Continuum flow regime: collisions between molecules are dominant and (b) Molecular flow regime: collisions between molecules and the walls of the channel. ... 6 Figure 2.5 - Hydrodynamic forces acting on particles in the flow field (Fext and Fcomp corresponds to forces along

extensional and compressional axis, respectively (Adapted from 38_{). ... 10}

Figure 2.6 - Schematic representation of E. coli. Flagella are helical filaments linked to the cell body that allow the rotation 41_{. ... 12}

Figure 2.7 - Mechanical effects on surfaces experienced by E. coli (Adapted from 43_{). ... 12}

Figure 3.1 – Schematic representation of the microdevice studied. (a) 3D configuration and (c) 2D configuration. 13 Figure 3.2 – Sketch of the bacteria. (a) Coccus bacteria and (b) Bacillus bacteria. ... 14 Figure 3.3 - 2D representation of the middle height plan of the microdevice model (a) Geometry boundaries: 1 - inlet of the feed channel, 2 – inlet of the waste channel, 3 – outlet of the feed channel, 4 – outlet of the waste channel. (b) Areas created for mesh refinement, including around the bacteria surface. ... 15 Figure 3.4 – (a) Representation of the mesh elements and close-ups of the different regions: (b) Microchannel with bacterium, (c) Microchannel with a bacillus bacterium. ... 17

Figure 4.1 - Representation of the rectangular channel used in the 2D simulations; dimensions and coordinate system associated (top view of the middle height plan). ... 19 Figure 4.2 - Comparison of analytical and numerical velocity profiles normalized with the average velocity in each case along different channel widths, at x = L, for aspect ratios (AR) equal to: (a) 0.05, (b) 0.075, (c) 0.15, (d) 0.30, (e) 0.50, and (f) 1.00. ... 20 Figure 4.3 - Velocity profiles along channel width for AR = 0.35. ... 21 Figure 4.4 - Velocity profiles along channel width for (a) AR = 0.3 and (b) AR = 2. ... 21 Figure 4.5 - 2D geometry with a schematic representation of the concentration field along streamwise direction. ... 22 Figure 4.6 – Top view of a device with a Y-junction inlet (with height H). ... 22 Figure 4.7 - Concentration maps at Y-junction inlet for Qw = 1 ml·min-1 and Qdw variable (from left to right:

0.25 ml·min-1_{, 1 ml·min}-1_{, and 4 ml·min}-1_{). From top to bottom: experimental results obtained by Sarkar et al.,}

numerical results obtained by Sarkar et al., and numerical results obtained in the present study. ... 23 Figure 4.8 - Location of the interface between two streams jointed by a Y-junction inlet, for different flow rate ratios: 0.25, 0.5, 1, 2, 4, 6, and 8. ... 23 Figure 4.9 - Analytical, experimental, and numerical concentration profiles along channel width at the outlet, in a device previously used by Holden and co-workers for different flow rates: (a) 500 nl·min-1_{, (b) 250 nl·min}-1_,

(c) 100 nl·min-1_{, and (d) 50 nl·min}-1_{. ... 24}

Figure 4.10 – Top view of a device with a T-junction inlet (with height H)... 25 Figure 4.11 – Diffusion thickness along channel width (abscissa and ordinate are in logarithmic scale), at the outlet, in a device previously used by Sarkar et al. (2014). ... 25

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Figure 4.12 - Geometry of the microdevice used for CFD simulations, (a) top view (x0y), and (b) side view (x0z) with the corresponding dimensions. ... 26 Figure 4.13 - (a) Concentration maps of glucose at t = 1 s for simulation results acquired in: (a) studies reported by Grünberger et al. and (b) in present project. ... 26 Figure 4.14 - Concentration maps of glucose at t = 6 s for simulating results acquired in: (a) studies reported by Grünberger et al. and (b) in present project. ... 26

Figure 5.1 – Velocity patterns in the first microchannel (M1): (1) coccus bacteria and (2) bacillus bacteria for a device similar to the base model (Lµ = 19.5 µm), for (a) VR = 1 and (b) VR =100. ... 28

Figure 5.2 – Concentration maps of glucose for a device similar to the base model (Lµ = 19.5 µm), for VR = 1, after:

(a) 0 ms, (b) 20 ms, (c) 60 ms, (d) 80 ms, (e) 120 ms, and (f) at steady-state. ... 29 Figure 5.3 - Concentration maps of lactic acid and close-up of the first (M1) and fifth (M5) microchannels for a device similar to the base model (Lµ = 19.5 µm), for VR = 1, after: (a) 0 ms, (b) 20 ms, (c) 60 ms, (d) 80 ms,

(e) 120 ms, and (f) at steady-state. ... 30 Figure 5.4 - Concentration maps of glucose for a device similar to the base model (Lµ = 19.5 µm), for VR = 100,

after: (a) 0 ms, (b) 20 ms, (c) 60 ms, (d) 80 ms, (e) 120 ms, (f) 400 ms, (g) 1000 ms, and (h) at steady-state. ... 31 Figure 5.5 – Concentration maps of lactic acid and close-ups of first (M1) and fifth (M5) microchannels for a device similar to the base model (Lµ = 19.5 µm), for VR = 100, after: (a) 0 ms, (b) 20 ms, (c) 60 ms, (d) 80 ms, (e) 120 ms,

(f) 400 ms, (g) 1000 ms, and (h) at steady-state. ... 31 Figure 5.6 - Concentrations along time, at the inlet and outlet of M1, and at the bacterium surface located in the middle of M1, for VR = 1: (a) Glucose and (b) Lactic acid. ... 32 Figure 5.7 – Concentrations along time, at the inlet and outlet of M1, and at the bacterium surface located in the middle of M1, for VR = 100: (a) Glucose and (b) Lactic acid. ... 32 Figure 5.8 - Schematic representation of the molar imbalances: (a) Boundaries where the calculations were done. Molar fluxes obtained for: (b) VR = 1 and (c) VR = 100. ... 32 Figure 5.9 - Concentration maps showing the effect of bacteria shape for a device similar to the base model (Lµ = 19.5 µm), for VR = 1: (a) Glucose concentration for coccus bacteria, (b) Glucose concentration for bacillus

bacteria, (c) Lactic acid concentration for coccus bacteria, (d) Lactic acid concentration for bacillus bacteria. ... 33 Figure 5.10 - Concentration maps showing the effect of bacteria shape for a device similar to the base model and (Lµ = 19.5 µm), for VR = 100: (a) Glucose concentration for coccus bacteria, (b) Glucose concentration for bacillus

bacteria, (c) Lactic acid concentration for coccus bacteria, (d) Lactic acid concentration for bacillus bacteria. ... 34 Figure 5.11 – VR effect on the concentrations for coccus bacteria (spherical), at the inlet and outlet of the first microchannel (M1) and at the bacterium surface located in the middle of the microchannel: (a) Glucose and (b) Lactic acid. ... 34 Figure 5.12 - VR effect on the concentrations for bacillus bacteria (rod-shaped), at the inlet and outlet of the first microchannel (M1) and at the bacterium surface located in the middle of the microchannel: (a) Glucose and (b) Lactic acid. ... 35 Figure 5.13 – VR effect on the lactic acid concentration for two different reaction stoichiometry, at the inlet and outlet of the first microchannel (M1) and at the bacterium surface located in the middle of the microchannel: (a) Coccus bacteria and (b) Bacillus bacteria. ... 35 Figure 5.14 – Flux maps for a device similar to the base model (Lµ = 19.5 µm), for VR = 1 and reaction

stoichiometry 1-2: (a) Glucose and (b) Lactic acid with the close-up for the first (M1) and fifth (M5) microchannels. ... 36 Figure 5.15 - Flux maps for a device similar to the base model (Lµ = 19.5 µm), for VR = 100 and reaction

stoichiometry 1-2: (a) Glucose and (b) Lactic acid with the close-up for the first (M1) and fifth (M5) microchannels. ... 36 Figure 5.16 – Concentration profiles at the bacteria surface, along x-direction for VR = 1 and reaction stoichiometry 1-2: (a) Glucose and (b) Lactic acid. Flux profiles at the bacteria surface, along x-direction: (c) Glucose and (d) Lactic acid. ... 36 Figure 5.17 - Concentration profiles at the bacteria surface, along x-direction for VR = 100 and reaction stoichiometry 1-2: (a) Glucose and (b) Lactic acid. Flux profiles at the bacteria surface, along x-direction: (c) Glucose and (d) Lactic acid. ... 37 Figure 5.18 – (a) Schematic representation of bacteria length increasing and (b) sketch of bacterial division. .... 37 Figure 5.19 – Glucose concentration maps (close-ups for M1 and M5) for a device similar to the base model (Lµ = 19.5 µm) for different bacteria lengths: (a) 1.25 µm, (b) 2.5 µm, and (c) 5.0 µm; VR = 1, reaction

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Figure 5.20 – Lactic acid concentration maps (close-ups for M1 and M5) for a device similar to the base model (Lµ = 19.5 µm) for different bacteria lengths: (a) 1.25 µm, (b) 2.5 µm, and (c) 5.0 µm; VR = 1, reaction

stoichiometry 1-2. ... 38 Figure 5.21 – Lactic acid concentration at the inlet and at the bacterium surface located at M1 microchannel, for different bacteria lengths; VR = 1 and reaction stoichiometry 1-2. ... 39 Figure 5.22 – Concentration at the inlet and outlet of the M1 microchannel, and at the bacterium surface located at the M1 microchannel, for different bacteria lengths; VR = 1 and reaction stoichiometry 1-2: (a) Glucose, (b) Lactic acid. ... 39 Figure 5.23 - (a) Velocity map (b) Glucose concentration map, (c) Glucose flux map, (d) Lactic acid concentration map, and (e) Lactic acid flux map and close-ups for M1 and M5 for a device similar to the base model (Lµ = 19.5 µm);

VR = 1 and reaction stoichiometry 1-2. ... 40 Figure 5.24 – (a) Velocity map, (b) Glucose concentration map, (c) Glucose flux map, (d) Lactic acid concentration map, and (e) Lactic acid flux map and close-ups for M1 and M5 for a device similar to the base model (Lµ = 19.5 µm);

VR = 100 and reaction stoichiometry 1-2. ... 40 Figure 5.25 – VR effect on the lactic acid concentration at inlet and at bacterium surface located in M1 microchannel with a portion outside the microchannel; reaction stoichiometry 1-2. ... 41 Figure 5.26 – VR effect on the glucose concentration for different kinetic models at the bacterium surface located at the middle of M1. ... 42 Figure 5.27 – Concentration maps for the kinetic model r = rmax for a device similar to the base model (Lµ = 19.5 µm);

VR = 1: (a) Glucose and (b) Lactic acid with close-ups for the first (M1) and fifth (M5) microchannels. ... 42 Figure 5.28 – VR effect on the concentration for the kinetic model r = rmax at the inlet and outlet of M1, and at the

bacterium surface located in the middle of M1: (a) Glucose and (b) Lactic acid. ... 42 Figure 5.29 – Concentration maps for the kinetic model r = c for a device similar to the base model (Lµ = 19.5 µm);

VR = 1: (a) Glucose and (b) Lactic acid with close-ups for the first (M1) and fifth (M5) microchannels. ... 43 Figure 5.30 – VR effect on the lactic acid concentration for the kinetic model r = c at the inlet and outlet of M1, and at the bacterium surface located at the middle of M1. ... 43 Figure 5.31 – VR effect on the concentration for the kinetic model r = c at the inlet (glucose and lactic acid) and outlet (lactic acid) of the M1 microchannel, and at the bacterium surface (lactic acid) located in the middle of the microchannel. ... 43 Figure 5.32 – Glucose concentration maps in devices with different microchannel lengths (Lm); for VR = 1 and

reaction stoichiometry 1-2: (a) 10 µm, (b) 20 µm, (c) 50 µm, (d) 70 µm, (e) 100 µm. ... 44 Figure 5.33 - Lactic acid concentration maps in devices with different microchannel lengths (Lm); for VR = 1 and

reaction stoichiometry 1-2: (a) 10 µm, (b) 20 µm, (c) 50 µm, (d) 70 µm, (e) 100 µm. ... 44 Figure 5.34 – Glucose concentration at the inlet and outlet of M1, and at the bacterium surface located in the middle of M1, for devices with different microchannels length; VR = 1 and reaction stoichiometry 1-2. ... 44 Figure 5.35 – Lactic acid concentration at the inlet and outlet of M1, and at the bacterium surface located in the middle of M1, for devices with different microchannels length; VR = 1 and reaction stoichiometry variable. ... 45

Figure C.1 - Mesh used during the simulations for the study of fluid flow. ... 57 Figure C.2 - Crops of the meshes used during the simulations for the study of fluid flow with close-ups: (1) 2D model and (2) 3D model. ... 57 Figure C.3 - Crops of the meshes used during the simulations for the study of mass transport and respective close-ups: (a) T-junction, (b) and (c) Y-junction, and (d) Microdevice with microchannels. ... 58 Figure C.4 – (a) Representation of the mesh elements and (b) Close-up of microchannels. ... 58

Figure D.1 - Concentration maps for a device similar to the base model (Lµ = 19.5 µm); VR = 1: (a) Glucose and

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List of Tables

Table 2.1 - Main fabrication methods with correspondent general description. ... 4

Table 2.2 - Dimensions of microdevice indicated above 21_{. ... 5}

Table 2.3 - Trapping techniques and main differences 13,14_{. ... 11}

Table 3.1 – Parameters of the microdevice model, its values and units. ... 14

Table 3.2 – Boundary-conditions known and imposed. ... 15

Table 3.3 – Parameter values for each parametric study analysed, comprising device, microchannels, bacteria, and surface reaction kinetics for 2D dimension. ... 16

Table 3.4 – Parameter values for the microdevice, fluid, and particle, throughout the simulations [meaning of these variables is indicated in Figures 3.1 and 3.2]. ... 16

Table 3.5 – Maximum elements size values used in the different zones for domain meshing. ... 17

Table 5.1 – Reaction kinetics studied and respective parameters. ... 41

Table B.1 - Simulation time, mass imbalance, and dimensionless numbers (Reynolds and Stokes) for feed and waste channels, and microchannel M1 for the respective studied parameter for different cases. ... 53

Table B.2 – Values of the inlet flow rates in the feed and waste channel, pressure drop in the feed channel, waste channel, and microchannel M1, and fluxes of consumed glucose and produced lactic acid in the model device. ... 55

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Notation and Glossary

A Cross-sectional area of the channel [m2_]

AR Aspect ratio of the channel [-]

ci Concentration of species i [mol·m-3]

Da Damköhler number [-]

Db Bacteria diameter [m]

Dh Hydraulic diameter of the channel (= 4A/Pw) [m]

Di Diffusion coefficient of species i in a solvent [m2·s-1]

H Device height [m]

Ki Substrate affinity [mol·m-3]

Kn Knudsen number [-]

L Channel length [m]

Lb Bacteria length [m]

Lin Length from the inlet to the 1st microchannel wall [m]

Lm Microchannels length [m]

Lout Length from the last microchannel wall to the outlet [m]

Lsp Space between microchannels [m]

Lµ Length with the microchannels [m]

ni Number of grid points [-]

Nµ Number of microchannels [-]

P Pressure [Pa]

Pe Péclet number [-]

Pw Wet perimeter [m]

rmax,i Maximum specific growth rate [mol·m-2·s-1]

Re Reynolds number [-]

St Stokes number [-]

t Time [s]

u Fluid velocity magnitude [m·s-1_]

𝑢⃗ Fluid velocity vector [m·s-1]

VR Velocity ratio (= uf/uw) [-]

W Channel width [m]

x,y,z Coordinate values in space [m]

Greek Letters

𝜌𝑓 Fluid density [kg·m-3]

𝜌𝑝 Particle density [kg·m-3]

𝜇𝑓 Fluid dynamic viscosity [Pa·s]

𝜂𝑓 Fluid kinematic viscosity [m2·s-1]

𝜏𝑓 Fluid characteristic time [s]

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Indexes in Inlet out Outlet c Channels f Feed channel m Microchannel w Waste channel

aver Average value

diff Diffusion mechanism

conv Convection mechanism

List of Acronyms

CFD Computational Fluid Dynamics

DNA Deoxyribonucleic acid

FEM Finite Element Method

LOC Lab-on-Chip

PDMS Polydimethylsiloxane PMMA Polymethylmethacrylate PTFE Polytrifluoroethylene

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Introduction 1

1 Introduction

1.1 Motivation

The main goal of the present project is the investigation by numerical simulation of a biocompatible microfluidic device with the ability to trap bacteria in a rapidly and sensitively way. This device should allow: 1. continuous supply of nutrients and continuous removal of metabolic products, outcome from glycolysis reaction in the surface of an E. coli bacteria trapped in a confined space; 2. avoid the overlap of the new cells formed; 3. monitoring the bacterial growth and consequent cellular division and; 4. observation of metabolic responses to environmental changes. This microdevice should be available for a wide range of applications, creating opportunities in a future manufacture.

Traditionally, the bacterial growth (lag, exponential, and stationary phases) can be carried out in two different ways: batch and continuous operations, each one with their advantages and disadvantages. The existing devices present limitations and so new microfluidic devices need to be developed.

In the past decade, considerable progresses have been made in microfluidics, a relatively new branch of science and technology, but with a high margin of progress. Many components of Lab-on-Chip systems can be designed and optimized using CFD (Computational Fluid Dynamics), where the predictive power of the simulation results is often high enough to draw reliable conclusions about the performance of components and devices.

1.2 SIMTECH Lab

Integrated in CEFT/FEUP (Transport Phenomena Research Centre) the SIMTECH Lab emerges as a research laboratory with a main focus on numerical techniques to model and simulate the transport phenomena in materials, products and systems at different scales and boundary conditions. In order to improve the existing solutions and/or respond to the actual problems the first focus are the testing and optimization stages, identifying the controlling mechanisms before the prototypes are built.

With a multidisciplinary team the SIMTECH Lab links different branches of science, Engineering, Biology, Physiology or Materials Science, promoting different fields of study: optimization of advanced microfluidic devices (BioTech), analysis of bacteria motion and mass exchange (MedTech, BioTech), integrated transport problems within and around the human body (MedTech), and smart solutions for body thermal protection (ProTech).

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Introduction 2

1.3 Thesis layout

The present thesis closes to the introductory part, where a general perspective of the motivation inherent to this work is presented. In part 2, Context and State-of-the-Art, the main information on microfluidic devices (applications, trapping strategies, chip design and fabrication methods, and flow patterns), as well as bacterial growth and continuum approach with the governing equations. The numerical method developed during this project is presented in part 3, Computational modeling and Device configuration, and validated in part 4, Validation.

In part 5, Results and Discussion, the results from the numerical simulations are presented, followed by their discussion. The main conclusions are discussed in the final part, Conclusions. The outlook of the thesis is offered at the final section of this part, including suggestions for future work.

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Context and State-of-the-Art 3

2 Context and State-of-the-Art

2.1 Micro-/nano- versus macroscale

Depending on several factors, including the apparatus size, at the micro-/nanoscale two approaches can be taken to describe the flow of a fluid: molecular and continuum approaches. A molecular approach implies the flow description of free molecules, while continuum assumes the matter in its indivisible form. The continuum approach was considered adequate for this work.

In the continuum approach, the main conservative equations describing the physics and chemistry of the flow processes in micro-/nanoscale are identical to those in macrofluidics. However, the mechanisms governing fluid flow, mass transport and reaction phenomena have different importance at different scales: inertial/viscous forces (Re), convection/diffusion contributions (Pe), Brownian motion (becomes more importance as particle size decreases - - individual particle-fluid collisions increase), particle motion (St), and influence of reaction in the mixing of the components (Da) 1,2_{. The scale reduction combined with higher surface-to-}

-volume ratios, induce short reaction times, high reaction efficiencies, and low material consumption 3_.

2.2 Microdevices applications

The development of microfluidic devices has become a popular research topic during the last 10-15 years, mainly provided by the advantages offered over conventional macroscale instruments, which includes: lower cost, lower sample and reagent consumption (and, consequently, lower reagent costs, faster reaction times 4_{, and even the achievement of}

multiple sample processing on a single device 5_{), adaptability for automation, and potential for}

portable point-of-care devices. Other advantages include elimination of time-consuming clinical testing in central laboratories, low unit cost of microfluidic structures for mass production, high throughput in parallel processing 6_{, and naturally ease-of-use}7_.

Besides that, microfluidics can be used to perform experimental procedures impossible at the macroscale 8_{; biological and chemical analysis are typically concerned with molecules}

and bioparticles with very small dimensions, which means that the tools used for their manipulation should be of a similar scale 9_{. The main areas where microdevices have been}

playing a role are medicine (e.g. diagnostic field 3,10,11_{), chemistry (e.g. chemical synthesis}12_),

and biology (e.g. cells, molecules and microparticles separation and manipulation in microfluidic systems 13,14_{; protein and biological pathogen analysis} 10_{; determination of}

genotype variations and different physiological states of the cells 15_{, and cell capture and}

counting, e.g. biological characterization) 9_.

Figure 2.1 shows different technologic applications and the corresponding biological target. Depending on the size of the biological organism, a technologic platform can be associated.

Other fields include industrial biotechnology, by introducing new ways to examine organisms 16_{, biomedical and environmental applications}11,17_{, pharmaceutics and food}12_,

aerospace industry and electronic devices 6_{, and biosensor fabrication}18_.

Trapping cells in microfluidic devices represents one motivation for the rapid increase of Lab-on-Chip research. With the expansion of advanced microfabrication techniques, methods to process cells on-line in microfluidic systems emerged as an important domain for the development of new experimental protocols.

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Context and State-of-the-Art 4 Figure 2.1 – Schematic representation of size comparison of biological targets and microfluidic technology

(Adapted from 19_).

2.3 Fabrication methods

Nowadays, different methods have been developed to fabricate microfluidic devices: soft lithography, micromachining, embossing, in situ construction, injection molding, and laser ablation 8_{. Each technique has their own advantages and disadvantages, depending on the}

specific applications of the device. Some of these methods are summarily described in Table 2.1, as well as their most important features and main differences.

Table 2.1 - Main fabrication methods with correspondent general description.

Fabrication method General description

Soft lithography 8,18 - Molding of a two-part polymer (elastomer and curing agent), called PDMS _{(polydimethylsiloxane), using photoresist masters.}

- This method is fast, less expensive, and suitable for the most biological applications. - Another similar fabrication method is photolithography; both methods require multiple

steps and, consequently, expensive procedures. Micromachining 8

- Connected with the recent advances in nanotechnology, is used to create manometer structures for microfluidic applications.

- Micromachining techniques are costly, intensive labor, and require highly specialized skills, equipment and facilities.

- The majority of the microfluidic applications do not require the precision that micromachining can offer.

Micromolding 8 _- _{Involves low cost fabrication, being a promising technique.}

- Limitations of injection molding for microfluidics include resolution and materials choice.

In situ construction 8_- _{New method developed recently, is carried out using photodefinable polymers.}

- This technique eliminates the bonding step needed in other methods (e.g. micromolding).

2.4 Chip design

The Lab-on-a-Chip main part consists of a PDMS structure which is manufactured using the technology of soft lithography 16_{. This technique responds to practical considerations (ease}

of fabrication and manufacture, and low cost) and at the same time to material properties (machinability, molecular adsorption, and optical properties). Addiction of PDMS, PMMA (polymethylmethacrylate), PTFE (polytrifluoroethylene), and quartz are the mainly used 6_.

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One field of science where the soft lithography is applied is Biology. In Figure 2.2 it is represented a microdevice fabricated using a standard lithography technique, capable of trapping bacteria. These bacteria enter in the channel jointly with the nutrients needed to guarantee their survival, being trapped into the microchannels.

(a) (b)

(c)

Figure 2.2 - (a) Schematic representation of the PDMS device with bound cover glass (Adapted from 20_{), (b) Top}

view of a microdevice example, and (c) 3D conformation, with the indication of main channel and microchannels, and the respectively inlet and outlet (Adapted from 21_).

The device consists of a series of microchannels, oriented at 90º angles to a trench through which growth medium flows at a constant rate 21_{. The length of the channel is chosen}

so as to ensure, by diffusion, sufficient supply of nutrients to the bacteria. This is a precise and powerful tool for microbial studies and can potentially be used to do research with different species of bacteria 20_{. The microdevices are constructed to restrict the bacteria grow in a simple}

monolayer and, consequently, to monitor single bacteria behaviour and growth in a fixed position 16_.

Table 2.2 - Dimensions of microdevice indicated above 21_.

Dimensions Value [µm]

Channel width, height ∼ 1

Channel length ∼ 25

Trench depth × width (not to scale) 25 × 100

Trench length 30

Other possible geometries using different trapping strategies [section 2.7] are presented in Figure 2.3. All of these configurations are complex and present fabrication limitations. In cases (a) to (c) different devices to trap single bacterium are observed, by using physical obstacles; in case (d) E. coli bacteria are trapped by optical forces. The conformations shown in cases (a) and (b) do not allow the removal of metabolic products, which is an essential feature of the present work, with consequent accumulation of non-reacting material and bacterial death in the microchannels.

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(a) (b) (c)

(d)

Figure 2.3 - Examples of cell trapping strategies in microfluidic devices. (a) Picolitre bioreactor for cultivation of bacteria; when the reactor is fully packed cells are pushed out of the overflow channels; design allows for

continuous cultivation and analysis (Adapted from 22_{). (b) Optical trapping in a photo-polymeric hydrogel}

(microfluidic device with three separate inlets; the red vertical arrow indicates the chemoeffector gradient1_{in the}

direction of the highest concentration) 23_{. (c) Hydrodynamic trapping using sieves (U-shaped with an aperture);}

streamlines are also represented, as well the red stagnation points (Adapted from 24_{). (d) Trapping bacteria}

through the optical force exerted on them along a microfluidic channel 25_.

2.5 Flow models in liquids: continuum versus molecular approach

The continuum model admits that the fluid is described as a continuous medium, i.e. described in terms of the spatial and temporal variations of the velocity, pressure, and others physical parameters. The conservative mass and momentum equations and the no-slip boundary condition at the wall are valid in space and time. In opposition, in free molecular flow regime in narrow spaces, the collisions between molecules can be neglected in comparison with the collisions of the molecules with the walls, i.e. the flow can be characterized by the displacement of each isolate molecule (Figure 2.4)1,26,27,28_.

(a) (b)

Figure 2.4 - Molecules in a parallel-plates channel. (a) Continuum flow regime: collisions between molecules are dominant and (b) Molecular flow regime: collisions between molecules and the walls of the channel.

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A simplest way to check the validity of the continuum approach is through the dimensionless Knudsen number (ratio between the mean free path of the molecules and the characteristic length of the apparatus (channel/microchannel diameter), Kn = 𝜆 𝑑⁄ ). Its calculation is only possible for gases. For liquids some authors presented conclusions from numerical and experimental data to validate the continuum approach:

 Samples volumes as little as 1 µl 29_.

 10 µm thickness contains 30 000 water molecules, enough to treat the flow to be in continuum 6_.

 Length scale of the flow larger than about 10 nm 27_.

 For water, when channel dimensions are above 1 µm 28,30_.

 The mean free path of the molecules is of the order of 0.2 times the diameter of the molecules. The continuum approach is still valid for length scales greater than about 10 molecular diameters (2-5 nm for water) 9_.

Based on the dimensions and/or conditions of the present study (aqueous solution of PBS and microchannels with a height of 1.5 µm) the continuum approach is assumed valid and representing in a realistic way the physical phenomena.

The governing equations in fluid dynamics for a continuum approach are introduced below 9_.

 Conservation of mass (continuity equation)

For an incompressible flow the continuity equation is written as:

∇. 𝑢⃗ = 0 (2.1)

where 𝑢⃗ is the fluid velocity vector.

 Conservation of momentum - Navier-Stokes equation prescribes solutions for fluid flow problems as long as initial and boundary conditions are specified. When the fluid has a Newtonian behaviour and the flow is incompressible Navier-Stokes equation is written as:

𝜌𝜕𝑢⃗

𝜕𝑡+ 𝜌 𝑢⃗ . ∇𝑢⃗ = −∇𝑝 + 𝜂 ∇

2_𝑢_⃗ (2.2)

where ∇𝑝 represents the pressure forces and 𝜂 ∇2_𝑢_{⃗ the viscous forces.}

In a steady pressure-driven flow through microchannels for Re ≪ 1, the inertial effects can be ignored and the viscous flow is called creeping (or inertialess) flow:

∇𝑝 = 𝜂 ∇2_𝑢_⃗ _(2.3)

The Reynolds number (Re) is usually defined as: Re =𝜌 𝑢𝑎𝑣𝑒𝑟 𝐷ℎ

𝜇 (2.4)

where 𝜌, 𝑢𝑎𝑣𝑒𝑟, 𝐷ℎ, and 𝜇 are, respectively, the fluid density [kg·m-3], the fluid average velocity

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 Conservation of energy

An important characteristic of microfluidic devices is the large surface area-to-volume ratio that allows a quick dissipation of the heat generated and, consequently, the flow can be assumed isothermal. The simulation temperature to be used attending to the E. coli growth is 37 ºC.

 Conservation of species

The mechanisms involved in the flow of species are diffusion and convection. In microdevices the species are diluted in the solvent (molar concentrations lower than 10% 31_),

meaning that Fick’s 1st_{law can be used to describe the diffusive transport}32_.

𝑗 𝑑𝑖𝑓𝑓,𝑖 = −𝐷𝑖∇𝑐𝑖 _(2.5)

where 𝑗 _{𝑑𝑖𝑓𝑓,𝑖} is the diffusive flux of specie 𝑖, 𝐷_𝑖 molecular diffusion coefficient and ∇𝑐_𝑖 the gradient concentration of specie 𝑖.

The Nernst-Planck equation (Fick’s 2nd_{law) combines diffusion (𝑗}

𝑑𝑖𝑓𝑓,𝑖= −𝐷𝑖∇𝑐𝑖) and

convection (𝑗 _{𝑐𝑜𝑛𝑣,𝑖}= 𝑢⃗ 𝑖 𝑐𝑖) mechanisms:

𝜕𝑐𝑖

𝜕𝑡 = −∇. [−𝐷𝑖 ∇𝑐𝑖+ 𝑢⃗ 𝑖 𝑐𝑖] (2.6)

Diffusive (𝑁_{𝑑𝑖𝑓𝑓}) and convective (𝑁_{𝑐𝑜𝑛𝑣}) contributions can be compared through the Péclet number (Pe):

Pe =𝑁𝑐𝑜𝑛𝑣 𝑁𝑑𝑖𝑓𝑓

=𝑢𝐿 𝐷

(2.7) where 𝑢 is the velocity magnitude [m·s-1_{], 𝐿 the characteristic length scale (size of the system)}

[m], and 𝐷 the characteristic molecular diffusion coefficient [m2_·s-1_].

Associated to the bacteria motion, the Stokes number (St) is widely used. This dimensionless number represents the ratio between the particle lag time (𝜏𝑝, characteristic

time of the bacteria) and the characteristic time over which the flow changes (𝜏_𝑓, characteristic time of the flow).

St =𝜏𝑝 𝜏𝑓 =𝜌𝑝𝑑𝑝 2_⁄_18𝜇 𝐷ℎ⁄𝑢𝑎𝑣𝑒𝑟 (2.8) where 𝜌𝑝 is the particle density and 𝑑𝑝 the bacteria equivalent diameter.

For low values of Stokes number (St ≪ 1), the particles follow the fluid streamlines (the drag forces associated to the particles are negligible), while for large Stokes number the particles path does not adjust to the flow streamlines.

Another term associated to the flow of species is the reaction term (𝑟_𝑖), which included in Eq. (2.6) gives:

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In the present work, the reaction term corresponds to the glycolysis reaction rate (reaction in the bacteria’s surface), which describes in a simplified way the bacterial metabolic process. The bacterial metabolic process is not simple and some important simplifications should be done to apply it into simulation modelling. In glycolysis reaction, two molecules of lactic acid (C3H6O3) are formed per each molecule of glucose (C6H12O6) consumed.

C6H12O6⟶ 2C3H6O3 _(2.10)

The Monod’ model was chosen to describe the bacterial growth. This kinetic model reports the specific growth rate (𝑟𝑖) as a function of the concentration of the growth-limiting

nutrient (glucose) (𝑐𝑖) and of two parameters (𝑟𝑚𝑎𝑥,𝑖 and 𝐾𝑖, the maximum specific growth rate

and the substrate affinity – Monod saturation constant respectively) 21,33,34_.

𝑟𝑖= 𝑟𝑚𝑎𝑥,𝑖

𝑐𝑖

𝑐𝑖+ 𝐾𝑖 (2.11)

The parameters are determined by Lineweaver-Burk linearization 35_:

1 𝑟𝑖 = 1 𝑟𝑚𝑎𝑥,𝑖 + 𝐾𝑖 𝑟𝑚𝑎𝑥,𝑖 1 𝑐𝑖 (2.12)

Other less established metabolic models have been studied to describe the bacterial growth, such as those developed by Herbert, Powell, Shehata and Marr, Blackman, Tessier, and Westerhoff et al., etc. 35,36_{. The growth kinetics information available is mostly limited to}

glucose, and few data have been published on kinetics with other nutrients (lactose, fructose, xylose …).

Then, Eq. (2.9) can be rewritten:

∇. (𝐷𝑖 ∇𝑐𝑖) + 𝑢⃗ . ∇𝑐𝑖 = 𝑟𝑚𝑎𝑥,𝑖

𝑐𝑖

𝑐𝑖+ 𝐾𝑖

(2.13)

The Damköhler number (Da) relates the competition between diffusion and reaction occurring in the microsystem 32,37_.

Da = 𝜏𝑑𝑖𝑓𝑓 𝜏𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛

(2.14) where 𝜏_{𝑑𝑖𝑓𝑓} defines the characteristic diffusion time and 𝜏_{𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛} the characteristic reaction time. One of two scenarios is expected: Da ≫ 1 (the reaction rate occurs faster than the diffusion) and Da ≪ 1 (the diffusion reaches the equilibrium for a low reaction time).

The flow in microfluidic devices is often in laminar regime (low Reynolds number) and minimal diffusion of species (high Péclet number), leading to slow mixing (high Damköhler number).

In addition to the previously covered topics, it is also necessary to discuss other subjects, including the flow patterns in microchannels, the trapping strategies and the bacterial growth/behavior.

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2.6 Flow patterns in microchannels

There are three different methods of transport in microdevices: 1. flow driven by

differential pressure; 2. electrokinetic flow (classified into four types: electrophoresis,

electro-osmosis, streaming potential, and sedimental potential) and; 3. capillary driving flow (flows owing to surface tension gradients) 9_{. In addition to flow across channels there is also}

flow through porous media and gels.

The transport occurs by means of applied pressure differences in a pressure-driven flow. The pressure gradient is maintained at the ends of the channel, causing flow to occur. At low Reynolds number, the flow is uniaxial along the channel/microchannel, and the velocity varies radially in a parabolic shape over the cross-section of the channel 6_{. Otherwise, in}

electro-osmotic the flow is initiated by the application of a high electric field.

In this study, the flow is pressure-driven and is modelled through the Navier-Stokes equations.

2.7 Trapping strategies

Trapping corresponds to the ability to restrict the movement of a cell in a controlled manner using microfluidics, allowing the study of individual cells and gain added insight into aspects of their physiology and behaviour 20_.

The trapping techniques can be regarded as separation methods applied to cells, particularly using hydrodynamics strategies 38_{(which can consist in a physical barrier, such as}

a sieve 4_{, or changes in the microchannels width, i.e. contraction-expansion array}

microchannel 7,39,40_{), dielectrophoretic techniques, optical gradients, magnetic and acoustic}

forces. Depending on the separation criteria, the methods are substantially different (Table 2.3) 14_.

Hydrodynamics method can be defined as the technique of realizing cell or particle trapping in microfluidic systems through the creation of side channels in a main transport channel. The side channels dimensions are sufficiently small to trap cells (i.e. bacteria)13_{. In}

an ideal trap, a positioned particle experiences zero net force and velocity (particle identified by 3 in Figure 2.5) 38_.

Figure 2.5 - Hydrodynamic forces acting on particles in the flow field (Fext and Fcomp corresponds to forces along

extensional and compressional axis, respectively (Adapted from 38_).

In dielectrophoresis, the forces are generated by a non-uniform electric field. In optical trapping, a tightly focused laser beam is used to trap and manipulate particles and cells with very high precision.

Magnetic trapping techniques utilize magnetic fields and magnetic particles of different kinds and sizes. Finally, acoustic trapping is mainly used for the handling of cells or bead agglomerates. In microfluidic systems the predominant applications of acoustic trapping are in cell studies, biosensors and, more recently, cell or particle enrichment 13_.

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Context and State-of-the-Art 11 Table 2.3 - Trapping techniques and main differences .

Trapping technique Hydrodynamic Dielectrophoresis Optical Magnetic Acoustic Discriminating

parameters 𝐷 𝐷, 𝜀 𝐷, 𝑛 𝐷, 𝜒 𝐷, 𝜌, 𝛽

Separation criteria NI Size, permittivity Size, shape,

refractive index susceptibility Size, compressibility Size, density,

Buffer demands NI pH, ion, clean

surfaces Transparent pH, ion, clean surfaces pH, ion, clean surfaces

System complexity

(non-contact mode) Low Medium Medium/high High Low

System complexity

(contact mode) Low Low operated in Scarcely contact mode

Low Scarcely operated in contact mode

Target sample NI Microparticles,

cells Microparticles, cells particles, Magnetic magnetically

tagged cells

Microparticles, cells

Trapping resolution Low (~10 µm)

Medium (~1 µm) High (~50 µm) Medium (~1 µm)

Low (~100 µm)

Single cell trapping Low Medium High Medium Low

Cell cluster trapping Medium Medium Low High High

Trapping force [pN] NA 200-400 100-2000 2-1000 100-400

Potential sources of lethal and non-lethal

effects on cells NI Joule heating and

electric field

Optical radiation and

temperature increase

Magnetic field Temperature increase

Instrumentation

requirement NI Fluidic pumps, and a function generator or DC

high voltage supply

Fluidic pumps

and laser optics Fluidic pumps and a permanent magnet Fluidic pumps, acoustic transducers, and a function generator or DC high voltage supply

Portability NI Portable Limited

portability due to optical setup

Highly

portable Portable

(Note 1: NA = Not applicable, NI = No information;

Note 2: D designs size, ε permittivity, n refractive index, χ susceptibility, ρ density, and β compressibility).

2.8 Bacterial growth/behavior (E. coli)

The most extensively studied bacterium is the bacteria E. coli. The small size and structural simplicity of E. coli (gram-negative bacterium, prokaryote) is an advantage complemented with the bacteria shape which is strikingly simple compared to those of higher eukaryotes. E. coli’s cellular body is rod-shaped, with an average length of ~ 1.5-5.5 µm, and thickness of ~ 1 µm (Figure 2.6) 41,42_{. The envelope of E. coli contains three layers: the}

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Context and State-of-the-Art 12 Figure 2.6 - Schematic representation of E. coli. Flagella are helical filaments linked to the cell body that allow

the rotation 41_.

The bacterial swimming occurs at very low Reynolds numbers (Re ≃ 10−4_{) and its motion}

is governed by Stokes flow (bacteria live in environments dominated by viscosity forces) 43,44_.

Bacteria are also subjected to surface-specific mechanics: adhesive forces, rheology of their surroundings, and transport rules which define their encounters with nutrients as shown in Figure 2.7.

Figure 2.7 - Mechanical effects on surfaces experienced by E. coli (Adapted from 43_).

Another important topic related to the bacterial behaviour and/or growth is the biofilms formation. A biofilm is nothing more than a complex microbial community embedded in a self--secreted matrix of polymeric substances, enhancing transport of nutrients and retention of suspended particles and, consequently, increasing the resistance to antimicrobial agents 45,46,47,48_{. These microbial communities are the major factor affecting the growth and}

spreading of bacteria 49_.

Bacteria survive and grow through detecting and, frequently, importing compounds present in their environments 43_{. In E. coli cultures, all cells are in the same steady state of}

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Computational modeling and Device configuration 13

3 Computational modeling and Device

configuration

3.1 Hardware and Software

The machine used in this study had 43 GB RAM and the simulations were performed in a FEM platform, using 10 processors in each simulation.

3.2 Device prototype

One of the final applications of this work is to assist the selection of the device configuration. The microdevice was not yet fabricated and no experiments were done.

Figure 3.1 shows the microdevice used during this work: 2D and 3D numerical simulations were performed. The device consists of two main channels and five orthogonal microchannels. One channel provides nutrients, the feed channel (left channel on Figure 3.1), and the other a carrier fluid to assist trapping and removal of secreted products, the waste channel (right channel on Figure 3.1).

(a) (b)

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Table 3.1 presents the parameters associated to the microdevice and the respective values and units.

Table 3.1 – Parameters of the microdevice model, its values and units.

Parameter Description Value Unit

H Device height 1.5 µm

Wf Feed channel width 30 µm

Ww Waste channel width 30 µm

Wm Microchannel width 1.5 µm

Lin Length from the inlet to the 1st microchannel wall 50 µm Lout Length from the last microchannel wall to the outlet 15 µm

Lsp Space between microchannels 3 µm

Lm Microchannel length 10 µm

Lµ Length with microchannels 19.5 µm

Db Bacteria diameter 1 µm

Lb Bacteria length 2.5 µm

Nµ Number of microchannels 5 -

A bacterium was considered to be positioned (i.e. trapped) in each microchannel, with its centre located in the geometric centre of the microchannels. Figure 3.2 shows the studied shapes.

(a) (b)

Figure 3.2 – Sketch of the bacteria: (a) Coccus bacteria and (b) Bacillus bacteria.

The Stokes flow in microchannels is often solved by numerical techniques, since experimental microfabrication techniques are not so well established, especially for complex geometries. Many components of Lab-on-Chip systems can be designed and optimized using computational simulations.

3.3 Species conditions

At the feed channel, the device is fed with a solution of glucose (nutrient that guarantees the bacterial growth and is consumed at their surface to produce lactic acid). An initial concentration of 200 mol·m-3_{was set, with diffusion coefficients of 5.4 × 10}-10_m2_·s-1_and

11.2 × 10-10_m2_·s-1_{for glucose [M}

glucose = 180.16 g·mol-1] and lactic acid [Mla = 90.08 g·mol-1],

respectively. The kinetic parameters assume the values 1.65 × 10-8_mol·s-1_·m-2_(r

max) and

4.5 mol·m-3_(K

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3.4 Geometry: Sub-domains and Boundaries

The device has two inlets marked with numbers 1 and 2 in Figure 3.3, respectively, inlet of the feed channel and inlet of the waste channel. The device outlets are marked with numbers 3 for the feed channel and 4 for the waste channel.

(a) (b)

Figure 3.3 - 2D representation of the middle height plan of the microdevice model (a) Geometry boundaries: 1 - inlet of the feed channel, 2 – inlet of the waste channel, 3 – outlet of the feed channel, 4 – outlet

of the waste channel. (b) Areas created for mesh refinement, including around the bacteria surface.

3.5 Physics interfaces and Boundary-conditions

The physic model chosen was Creeping Flow (Single Phase Flow) (spf) for the fluid flow study. It is adequate for microfluidics (Re << 1), where the convective term in the Navier-Stokes equation can be dropped. In order to describe the mass transport, the interface adopted was Transport of Diluted Species (tds); this model is indicated for dilute mixtures. Table 3.2 indicates the boundary-conditions for the two models (fluid flow and mass transport) 9_.

Table 3.2 – Boundary-conditions known and imposed.

Fluid flow Mass transport

- No-slip condition at the walls, 𝑢|𝑤𝑎𝑙𝑙= 0

(dynamic boundary condition for continuity of tangential velocity)2_.

- Shallow channel approximation. - ρf = 103 kg·m-3 and µf = 10-3 Pa·s-1.

- uf = uw = 1 mm·s-1.

- Outlet: Pf = Pw = 0.

- Dglucose = 5.4 × 10-10 m2·s-1 and Dlactic acid = 11.2 × 10-10 m2·s-1.

- C0,glucose = 200 mol·m-3.

- Species fluxes at the bacteria surface: Inward flux (𝑁0,𝑖)

specifies the flux of each species 𝑖 individually representing the flux due to reaction. The flux of glucose leaves the system, a minus sign should be used.

𝑁0,𝑖= −𝑛. 𝑁𝑖= −𝑛(−𝐷𝑖 ∇𝑐𝑖+ 𝑢⃗ 𝑖 𝑐𝑖)

- Outflow: Feed and waste channels.

2_{PDMS surface undergo surface treatments (i.e. plasma oxidation), that transforms the hydrophobic surface in a}

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3.6 Parametric analysis

The major goal of this study is to investigate the fluid flow and mass transport in a microfluidic device for trapping bacteria, to study the parameters affecting the bacteria exposure to nutrients (glucose) and the removal of their metabolic products (lactic acid). The studied parameters are shown in Table 3.3. Table 3.4 shows the parameter used throughout the numerical simulations, except where mentioned otherwise.

Table 3.3 – Parameter values for each parametric study analysed, comprising device, microchannels, bacteria, and surface reaction kinetics for 2D dimension.

Case

Device µchannels Bacteria Surface reaction kinetics

VR = uf/uw [-] Hc [µm] Lm [µm] Shape3 _L_b [µm] n Reaction kinetics rmax K [mol·m-3_] I 100|50|10|5|1 1.5 10 Spherical 2.5 1 Monod 1.65 × 10-8 [mol·s-1_·m-2_] 4.5 II 100|50|10|5|1 1.5 10 Spherical 2.5 2 Monod 1.65 × 10-8 [mol·s-1_·m-2_] 4.5 III 100|50|10|5|1 1.5 10 Rod- -shaped 2.5 1 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5 IV 100|50|10|5|1 1.5 10 Rod- -shaped 2.5 2 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5 V 1 1.5 10|20|50|70|100 Rod- -shaped 2.5 1 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5 VI 1 1.5 10|20|50|70|100 Rod- -shaped 2.5 2 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5 VII 1 1.5 10 Rod- -shaped 2.5|3.75|5|7.5 2 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5 VIII 100|50|10|5|1 1.5 10 Rod- -shaped 7.5 2 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5 IX 100|50|10|5|1 1.5 10 Rod- -shaped 2.5 2 𝑟1= 𝑟𝑚𝑎𝑥 1.65 × 10 -8 [mol·s-1_·m-2_] - X 100|50|10|5|1 1.5 10 Rod- -shaped 2.5 2 𝑟2= 𝑟𝑚𝑎𝑥𝑐 1 [m·s -1_] _- XI 100|50|10|5|1 1.5 10 Rod- -shaped 2.5 2 𝑟3= 𝑐 𝐾 + 𝑐 - 4.5 XII 100|50|10|5|1 1.5 10 Rod- -shaped 2.5 2 𝑟4= 𝑟𝑚𝑎𝑥 𝑐 𝐾 1.65 × 10-8 [mol·s-1_·m-2_] 10 3 XIII 100|1 1.5 10 Rod- -shaped 2.5 2 Monod 1.65 × 10 -8 [mol·s-1_·m-2_] 4.5

Table 3.4 – Parameter values for the microdevice, fluid, and particle, throughout the simulations [meaning of these variables is indicated in Figures 3.1 and 3.2].

Device Fluid Bacteria

Lin [µm] Lout [µm] Ww [µm] Wm [µm] ρf [kg·m-3] µf [Pa·s-1] ρp [kg·m-3] Db [µm] Lb [µm]

50 15 30 1.5 1 000 0.001 1 040 1 2.5

3_{A coccus is any bacterium that has a spherical shape. Another distinct bacterial shape is the rod-shaped, a bacillus}

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3.7 Mesh

The software uses a mesh grid to discretize the equations, applying the finite element method (FEM software) to numerically solve the partial differential equations. An unstructured mesh with triangular elements was employed, with increased density in the vicinity of the channel walls (Figure 3.4).

(a) (b)

(c)

Figure 3.4 – (a) Representation of the mesh elements and close-ups of the different regions: (b) Microchannel with bacterium, (c) Microchannel with a bacillus bacterium.

The geometry was divided in several areas according to the refinement need (Figure 3.3b). In each zone, the maximum element size is controlled by Wf, Ww, Wm, nf, nw, and

nm (Table 3.5, nf, nw, and nm represent the minimum number of points created by the grid

elements; in each area they were kept constant and equal to 15).

Table 3.5 – Maximum elements size values used in the different zones for domain meshing. Mesh refinement zones Maximum element size

Area a 2 × Wm/nm

Area b1 Wf/nf

Area b2 Ww/nw

Area c 1.5 × Wm/nm

Area d Wm/nm

Mesh refinement zones Maximum element size

Area e 0.025

Feed channel walls 0.95 × Wf/nf

Waste channel walls 0.95 × Ww/nw

Microchannels walls 0.95 × Wm/nm

In order to check the conservation of mass, the values of mass imbalance were determined and are indicated in Appendix B. Reynolds and Stokes numbers were also monitored for the main channels and for the first microchannel (M1) to ensure the applicability of the model simplification4_{. The expressions used for the calculations are presented in Appendix A.}

4_{Particle motion: Stokes number in the simulations is much lower than 1, which means that the particle trajectory}

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3.8 Discretization and convergence criteria

The discretization consists in the division of a continuous system into a finite number of elements with finite size. In fluid flow, the discretization of two components, velocity and pressure, is required, for which second order elements and linear elements (first order), where used. The element order chosen for mass transport was quadratic (second order) 50_{. The relative}

tolerance for fluid flow, pressure and concentration calculations was set to 10-9_.

Additionally, values of mass imbalance were determined to ensure the conservation in each simulation. Reynolds number and Stokes number were monitored for the main channels and for the first microchannel (M1) to check the applicability of the model simplification for the flow, in all the simulations.

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Validation 19

4 Validation

The main goal of this thesis is to obtain, by numerical simulation, the necessary knowhow to the construction of a versatile microfluidic device to trap and feed bacteria. In this way, the numerical procedure used must be validated, using for that purpose, supported experimental, analytical and numerical data as benchmark.

A finite element analysis package was used to simulate the transport phenomena involved: fluid flow5_{, transport of nutrients in a continuous flow}6_{, and surface reaction}7

(glycolysis). For that purpose, it is necessary to solve, simultaneously, the Navier-Stokes equation and time dependent convection-diffusion equation (Fick’s 2nd_{law). Images of the}

tested meshes and close-ups are indicated in Appendix C.

1. Fluid flow

The first step is the validation of the fluid flow. A numerical study on laminar pressure - driven flow through a microchannel (geometric element of the microdevice) was done. Using the FEM platform, velocity field calculations were done for creeping flow (Stokes flow) in a rectangular shallow channel.

Figure 4.1 - Representation of the rectangular channel used in the 2D simulations; dimensions and coordinate system associated (top view of the middle height plan).

The dimensions L and W in Figure 4.1 are, respectively, the length and the width of the channel. The height H is not represented.

An important dimensionless parameter is the aspect ratio – AR -, which corresponds to the height (H) to width (W) ratio. The numerical simulations were conducted for several devices with constant height (H = 1.5 µm) and decreasing width, so that AR assumed different values: 0.05, 0.075, 0.15, 0.3, 0.5 and 1.

The obtained data (2D) were compared with the results from a 3D analytical correlation (Eq. (4.1)) (truncated at the third term (i = 5)) for fluid flow in rectangular ducts 27_{, and are}

presented in Figure 4.28,9_. 𝑢𝑥(𝑦, 𝑧) = 4𝐻2_∆𝑃 𝜋3_𝜇𝐿 ∑ 1 𝑛3 ∞ 𝑛=1,3,5,… [1 − 𝑐𝑜𝑠ℎ (𝑛𝜋 𝑦 𝐻) 𝑐𝑜𝑠ℎ (𝑛𝜋_2𝐻)𝑊 ] 𝑠𝑖𝑛 (𝑛𝜋𝑧 𝐻) (4.1)

5_{Location folder: X:\PROJECTS\Proj08 - MICROFLOW 2.0\0. Validation\1. Flow}

6_{Location folder: X:\PROJECTS\Proj08 - MICROFLOW 2.0\0. Validation\2. Mass transport} 7_{Location folder: X:\PROJECTS\Proj08 - MICROFLOW 2.0\0. Validation\3. Surface reaction}

8_{Excel file: X:\PROJECTS\Proj08 - MICROFLOW 2.0\0. Validation\1. Flow\Images & Others\nguyen_AR_study.xlsx} 9_{Numerical model: X:\PROJECTS\Proj08 - MICROFLOW 2.0\0. Validation\1. Flow\AR_study.mph}

Numerical analysis of transport processes in a microfluidic device for bacteria confinement