Transport Phenomena - DCU

Contents

1. Introduction

9. Liquid Handling

2. Fluids

10.Microarrays

3. Physics of Microfluidic Systems

11.Microreactors 12.Analytical Chips

4. Microfabrication Technologies Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

13.Particle-Laden Fluids

5. Flow Control

a. Measurement Techniques

6. Micropumps 7. Sensors

b. Fundamentals of Biotechnology

8. Ink-Jet Technology

c. High-Throughput Screening

Microfluidics - Jens Ducrée

Fluids: Transport Phenomena

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2. Fluids 2.1. General Characteristics 2.2. Dispersions 2.3. Thermodynamics 2.4. Transport Phenomena 2.5. Solutions 2.6. Surface Tension Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

2.7. Electrical Properties 2.8. Optical Properties 2.9. Biological Fluids



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2.4. Transport Phenomena 2.4.1. Diffusion 2.4.2. Viscosity 2.4.3. Transport of Heat 2.4.4. Characteristic Numbers

Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el



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2.4 Summary of Phenomenological Laws  Transport processes  Arbitrary thermal motion on molecular level  Driven by gradients (inhomogeneities)  Mostly linear relationship Flow = coefficient  force




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2.4.1. Diffusion

 Diffusion  Counteracts nonuniform particle densities  Thermal „Brownian“ motion  Process underlying all other transport phenomena in fluids

 Fick‘s first law


 Current density jN antiparallel to gradient  Systems seeks homogeneity  Diffusion coefficient D [m² s-1]



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2.4.1. Diffusion  Fick‘s second law  Time domain  Fixed location

 Combining Fick‘s first law and equation of continuity


 Laplace equation

 Stationary conditions



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2.4.1 Molecular Picture  Derivation of diffusion constant D  Flow through surface z1  Between two planes with diverging particle densities  Distance = mean free path lmfp  Particles reach plane without collisions (statistically) Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Linear term of Taylor expansion Microfluidics - Jens Ducrée


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2.4.1 Diffusion coefficient  Calculation of diffusion coefficient  Assumption: uniformly distributed directions of all vectors v

average over Ensemble




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2.4.1 Examples for Diffusional Processes  Example  Bilateral diffusion from 2-D layer  Within pure solvent  Initial conditions


 Solution by „educated guess“



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2.4.1 Examples for Diffusive Processes




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2.4.1 Examples for Diffusive Processes  Example  Diffusion through permeable diaphragm  Initial conditions




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2.4.1. Laminar Mixing




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2.4.1 Time and Length Scales  Distance

2 (“rules of thumb”)

 Time


2 ~l2/D



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2.4.1 Brownian Motion  Discovered by Scottish Botanist Robert Brown in 1827  Random thermal motion of pollen under microscope

 Closely related to thermal velocity


 Mesoscopic particle    

Width of distribution of particle locations

r0 = 1 µm m = 10-15 kg T = 293 K vT  3 mm s-1



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2.4.1 Brownian Motion

 Fluorescent particles Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 4 seconds of data  2 µm in diameter  Left picture  Particles moving in pure water

 Right picture shows  Particles moving in concentrated solution of DNA  I.e., in viscoelastic solution in other words From http://www.deas.harvard.edu/projects/weitzlab/research/brownian.html



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2.4.1 Brownian Motion


 Erratic motion of single milk droplets in water  Droplets size of about 1 micrometer  Continuously kicked by fast-moving water molecules  Size 5000 times smaller

 Magnification factor of about 10,000 http://www.microscopy-uk.org.uk/



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2.4.1 Boltzmann Distribution  System at T in thermal equilibrium  Each particle with certain kinetic energy  Potential gradient  Particles accumulate in potential minimum

 Counteracting diffusive current


 Boltzmann distribution  Ratio of particle densities Ni at two locations separated by potential difference E Boltzmann function at T = 273 K calibrated to N0 = 1



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2.4.1 Maxwell Distribution  Maxwell distribution  Fraction of molecules in velocity interval [v, v + dv]  f(v)dv is probability of finding a vector v in [v, v + dv]  Normalization of integrand function f(v) to unity Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Leads to Maxwell distribution



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2.4.1 Maxwell Distribution  Transformation into energy space  Replacing v with kinetic energy E = 1/2 mv2


 Helps determining which fraction of particles is capable of „jumping“ over a certain potential energy barrier E0  Analytical expression for this fraction



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2.4.1 Reaction Kinetics  Chemical Reactions  Binary collisions  Collision rate scales linearly with vT and thus with T1/2  Concentration of reaction partners c also influences speed of reaction  For single step reaction


 Reaction rate

- Governed by stoichiometry (x, y, ...)  Concentrations c(X), c(Y), …  Reaction velocity constant k’c Microfluidics - Jens Ducrée


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2.4.1 Reaction Kinetics  Chemical reactions  Activation energy (Gibbs energy)  Simple model reaction

 For simplicity: G = U = Eact  Typically Eact = 60 to 250 kJ mol-1  G = Gact – G‘act Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 G < 0 exothermic  G > 0 endothermic

 Velocity constant of reaction  Strongly dependent on T  Reaction-specific  Arrhenius equation



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2.4.2. Viscosity  Viscosity  Transfer of momentum from one plane sliding parallel to another  Mediated by fluid sandwiched between them  „Internal friction“ of fluid

 Newton’s law of viscosity Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Relates flow of axial momentum pz = m vZ along lateral x-direction from one plane sliding parallel to another one by viscosity   Unit of : Pa s = kg m-1 s-1 or old Poise with 1 Ps = 0.1 Pa s



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2.4.2 Viscosity of Gases  Assumptions for picture: - Flow in z-direction - Uniform particle density

 Net flux in x-direction of longitudinal momentum pz

Newton Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Viscosity of gases

T

 Kinematic viscosity - „Momentum diffusivity“



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2.4.2 Viscosity of Gases

Order: 10-5 Pa s Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el



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2.4.2 Viscosity of Liquids  Liquid  Very dense gas  lmfp  average distance between molecules  Macroscopic behavior governed by intermolecular potentials

 Picture  Moving A to A‘  Activation energy E0  External shear stress xz


- Force f on each particle on wall - f counteracts motion in negative z-direction - Momentum loss transferred to B - „Activation energy“ E = f a / 2



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2.4.2 Viscosity of Liquids  Transfer of momentum pZ = m vZ  Lateral velocity gradient

 net difference between number of jumps +Z and -Z in positive and negative z-direction, respectively  Evaluation of gradient (using Boltzmann Ansatz) Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

0 fundamental oscillation frequency of molecule between neighbors Microfluidics - Jens Ducrée


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2.4.2 Viscosity of Liquids  First order Taylor expansion for E / kBT

 Viscosity of liquids Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

T



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2.4.2 Viscosity of Liquids

Order: 10-3 Pa s (gases: 10-5 Pa s)




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2.4.2 Newtonian Fluids  Newtonian fluids  Classical idealized fluids  Linear stress-strain relationship  I.e. linear relationship between tension tensor  and tensor of rate of deformation (strain rate)  - Viscosity  : constant of proportionality

 Approximated by gases and „simple“ liquids like water


 Non-Newtonian fluids  Dispersions containing large structured macromolecules  E.g. polymers and proteins



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2.4.3. Transport of Heat  Equilibrium thermodynamics  System seeks uniform temperature distribution

 Inhomogeneous T-distribution  Transport of heat

 Two basic modes  Diffusion - Statistical phenomenon (related to entropy) - Summarized in macroscopic thermal conductivity  Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Convection - Far more complicated - Macroscopic ramifications - Minor importance in microworld

 Another frequently encountered issue  Transfer of thermal energy at interface between two media/phases  No inter-phase diffusion / exchange of particles  Transmission and transition of heat Microfluidics - Jens Ducrée


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2.4.3 Thermal Conductivity  Fourier‘s law  Basic equation quantifying diffusive transport of heat

 Net flow of energy jQ  Thermal conductivity 

 Power


 Cross-section of flow A

 Relaxation time for temperature gradient  Distance d Microfluidics - Jens Ducrée


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2.4.3 Thermal Conductivity  Molecular picture  Net energy flow in positive z-direction  Using linear terms of Taylor-expansion




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2.4.3 Thermal Conductivity  Thermal conductivity

 Approximating on lmfp


 Independent of particle density N for ideal gas

 Heat diffusion coefficient kinematic viscosity



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2.4.3 Thermal Conductivity  Typical values




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2.4.3 Convection

 Macroscopic transport of particles  Characteristic for macro-devices  Important for heat transport in liquids and gases

 Forced convection  Transport of heat by macroscopic particle flow  E.g., by an external mechanical source Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Free convection  For instance driven by buoyancy



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2.4.3 Convection

 Convective currents  Stationary or non-stationary flow patterns

 Simulation of convection  Often fail to coincide with experimental observations  E.g., atmospheric weather and climate Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Free Convection of minor importance in microdevices  Laminar conditions Microfluidics - Jens Ducrée


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2.4.4. Characteristic Numbers  Fourier mass number  Characterizes diffusion

 Time t, e.g. residence in chemical reaction chamber  Typical diffusive time scale tD = l2 / D Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Schmidt number  Relates viscosity and diffusion

 Roughly 0.8 for gases



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2.4.4. Characteristic Numbers  Fourier number  Diffusion of heat  Stored thermal energy

 Prandtl number


 Ratio between momentum diffusivity (via dynamic viscosity ) and heat diffusivity (via thermal conductivity )

 Specific heat capacity Cm  Typically 3 to 300 for liquids and 0.7 to1.0 for gases Microfluidics - Jens Ducrée


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Transport Phenomena - DCU

Transport Phenomena - DCU

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